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Chef and Periodic Sequence

monk616 (1)
What is the logic to solve this problem?

Let's define a periodic infinite sequence S (0-indexed) with period K using the formula Si=(i%K)+1.

Chef has found a sequence of positive integers A with length N buried underground. He suspects that it is a contiguous subsequence of some periodic sequence. Unfortunately, some elements of A are unreadable. Can you tell Chef the longest possible period K of an infinite periodic sequence which contains A (after suitably filling in the unreadable elements) as a contiguous subsequence?
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