Let \(f\) be a function from the non-negative integers to the positive integers with \(f(0) = f(1) = 1\) and
\[\frac{f(x)f(y+1)}{f(y)} + \frac{f(x)f(x-y)}{f(x-y-1)} = f(x+1).\]
Find the exponent of 3 in the prime factorization of \(f(2016)\).

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