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Let $S$ be the set of all nonzero real numbers. Let $f : S \to S$ be a function such that $f(x) + f(y) = f(xyf(x + y))$for all $x,$ $y \in S$ such that $x + y \neq 0.$  Let $n$ be the number of possible values of $f(4),$ and let $s$ be the sum of all possible values of $f(4).$ Find $n \times s.$

May 6, 2021