# Many solutions

Level pending

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.

×