<human>: I asked this girl out just over a month ago, and got rejected in a friendly manner. Reason for the rejection was that she was in a relationship. We've texted before, but our friendship was pretty impersonal and new. We continued texting over short periods of time after the rejection, I had no problem with it.

About a week ago, she personally told one of my friends that she's single and I should ask her out. So I did. We went on two lunch dates which went okay in my opinion.

Today she reveals to me that it's all too soon for her, considering her recent break up from an apparent obsessive boyfriend. She considers me sweet (not the best thing to say obviously, but I can live), and wants to remain friends and see how it goes from there.

I understand her feelings from my own experience, but man I really feel shitty after being rejected, dating twice and rejected again. I'm interested in this girl, but not so much that leaving this behind isn't an option. Would it be reasonable to continue a friendship that consists of date-like activities without the "romance"? I'd prefer keeping our new relationship, but I'm open to ideas.

Can you provide a brief rundown of the text in within 3 sentences?
<bot>: Recently broken up girl believes that our new relationship is too fast for her. I want to keep the relationship, but I'm open to friendship. What should I do?