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Let \mathbb{Z}_{>0} denote the set of positive integers. Consider a function f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}. For any m, n \in \mathbb{Z}_{>0} we write f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots)). Suppose that f has the following two properties:

(i) if m, n \in \mathbb{Z}_{>0}, then \frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0};

(ii) The set \mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\} is finite.

Prove that the sequence f(1) - 1, f(2) - 2, f(3) - 3, \ldots is periodic.


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