Weird Functional Equation

Algebra Level 4

\[f^{f(a)}(b) f^{f(b)}(a) = \big(f(a+b)\big)^2\]

Let \(f:\mathbb{N} \to \mathbb{N}\) be an injective function such that the above holds true for all \(a,b \in \mathbb{N}\). Let \(S\) be the sum of all possible values of \(f(2017)\). Find \(S \bmod{1000}\).

\(\)
Note: \(f^{k} (n)\) means \(\underbrace{f\big(f(f(\ldots f(n)\ldots))\big)}_{\text{number of }f \text{'s}\ = \ k}.\)

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