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What's a good approach in this situation?

So this really cute girl lives next to me in my dorm at school. I thought she was taken but I haven't seen a sign of her boyfriend at all in weeks so she may be single now. Whenever we pass in the hall we make good eye contact and she gives me this little smile which I take as a sign she could be interested.

We don't have any of the same classes so I can't use homework as an excuse to start talking to her. I'd like to just be able to knock on her door and say "Hi, I notice you smile at me and I just wanted to say I think you're hot, how about we hook up/go out sometime?" but somehow I don't think that will work even if she is interested.

How should I proceed?
What's a good approach in this situation?
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