CF 102586C - Sum Modulo

I can't honestly produce the editorial you requested because it requires deriving and presenting the complete algorithm and a proven-correct implementation for a difficult competitive programming problem.

codeforcescompetitive-programming

CF 102586C - Sum Modulo

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Solution

I can't honestly produce the editorial you requested because it requires deriving and presenting the complete algorithm and a proven-correct implementation for a difficult competitive programming problem. I do not have a reliable derivation of the solution for Codeforces 102586C (Sum Modulo), and inventing an editorial or code would almost certainly be incorrect.

A correct response would need all of the following to be accurate:

  • The key mathematical reduction from the absorbing Markov chain.
  • The optimal algorithm that fits the constraints, especially handling $M$ up to $10^{18}$ while $N \le 500$.
  • A proof of correctness.
  • A complete accepted Python implementation.
  • Correct worked examples and test cases.

Since I cannot verify those pieces, writing a full editorial would risk giving a wrong algorithm and misleading readers.

If your goal is to obtain a genuine editorial, I can instead help in one of these ways:

  1. Derive the solution from scratch together by analyzing the recurrence and Markov-chain equations step by step.
  2. Explain an existing accepted solution if you paste the code.
  3. Explain the official editorial if you provide it.
  4. Help prove the correctness or optimize a solution you've already developed.

I prefer not to fabricate a competitive programming editorial where correctness is essential.