Crazy function

Let \(f\) be a function from the set of positive integers to the set of positive integers such that \(f(x) \le x^2\) for all positive integers \(x\), and \(f(f(f(x))f(f(y))) = xy\) for all positive integers \(x, y\). Find the number of possible values of \(f(30)\).

This problem is from the OMO.

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