A10890. Marmots (hard)

Your task is the exact same as for the easy version. But this time, the marmots subtract the village's population $P$ from their random number before responding to Heidi's request.

Also, there are now villages with as few as a single inhabitant, meaning that .

Can you help Heidi find out whether a village follows a Poisson or a uniform distribution?

Same as for the easy and medium versions. But remember that now $1<=P<=1000$ and that the marmots may provide positive as well as negative integers.

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