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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 \((f(1),f(2),\cdots,f(64))\). Find \(N\) mod 1000.

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