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I might have to take an incomplete for one of my classes this semester. I don't know if my school reports it as a punitive or non-punitive grade.

Someone tell me if my following plan will work: I take the incomplete, finish the work during break, submit it ASAP, and have my transcript updated before I send in fall grades to LSAC OR

bite the bullet and try to get all my shit done, even though it might kill me. I've already submitted all my apps. Will schools reach a decision before the middle of January, when spring semester starts?

Yea they'll reach a decision, but schools will often ask for your final transcripts/proof of graduation. If you have an incomplete, you may not graduate.