To collect one's thoughts means to take time for his/herself after a particularly disturbing event, and to get his/her mind back in order.
Person1: Hey, what's up?
Person2: Becky and I just broke up last night.
Person1: Oh that sucks, how've you been handling it?
Person2: I dunno man, but I feel like complete shit.
Person1: I'm sorry, how about I leave you alone and give you time to collect your thoughts?
Person2: Good idea. I'll explain things later then.
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To know something `offhand`: without having to refer to a source.
PersonA: Hey, do you by chance know offhand who's going to be teaching physics this semester?
PersonB: No sorry, I didn't check.
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