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Well, it seems to me that you're just over-complicating matters by considering it to be a "FB chat conversation".
There are no special protocols for online interactions, as compared to say, text interactions or even , in some cases, real time interactions.
Keep the conversation light, fun but don't show too much interest, too soon. Treat her like a girl you're gaming in real life, except when it comes to online interactions you need to focus on building comfort quickly, otherwise, she could easily dismiss you as one of the hundred odd guys who send her a new message a month.
Either way, give time and efforts, and go for her phone number with as fast a time limit as you can, but remember to develop enough attraction and comfort before pulling that trigger.
Good luck.
Excellent Advice Don! I'm being saying this for years.
