CF802F 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

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