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For any integer n\geq 2, we compute the integer h(n) by applying the following procedure to its decimal representation. Let r be the rightmost digit of n.
If r=0, then the decimal representation of h(n) results from the decimal representation of n by removing this rightmost digit 0.If 1\leq r \leq 9 we split the decimal representation of n into a maximal right part R that solely consists of digits not less than r and into a left part L that either is empty or ends with a digit strictly smaller than r. Then the decimal representation of h(n) consists of the decimal representation of L, followed by two copies of the decimal representation of R-1. For instance, for the number 17,151,345,543, we will have L=17,151, R=345,543 and h(n)=17,151,345,542,345,542.Prove that, starting with an arbitrary integer n\geq 2, iterated application of h produces the integer 1 after finitely many steps.

Proposed by Gerhard Woeginger, Austria

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