The positive integer-valued function \(f(n)\) satisfies \(f(f(n)) = 4n\) and \(f(n + 1) > f(n) > 0\) for all positive integers \(n\). Let \(N\) be the number of possible 64-tuples \((f(1), f(2), f(3), \dots, f(64))\). Find \(N\) mod 1000.

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