You wish they were multiplied

\[ \displaystyle f(n+2)=\frac { f(n+1)+f(n) }{ \gcd(f(n),f(n+1)) }\]

A function \(f : \mathbb{N} \to \mathbb{N}\) satisfies the above equation for all \(n \in \mathbb{N}\) and it has an upper bound. Find the sum of all possible values of \(f(2016)\).

Clarification: \(\mathbb{N}\) denotes the set of positive integers \(\{1, 2, 3, \ldots\}\).

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