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For positive integers n, the numbers f(n) are defined inductively as follows: f(1) = 1, and for every positive integer n, f(n+1) is the greatest integer m such that there is an arithmetic progression of positive integers a_1 < a_2 < \ldots < a_m = n for which

f(a_1) = f(a_2) = \ldots = f(a_m).

Prove that there are positive integers a and b such that f(an+b) = n+2 for every positive integer n.

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