NMC 2003 Problem 4

Let $\mathbb{R}^{*}=\mathbb{R} \backslash\{0\}$ be the set of non-zero real numbers. Find all functions $f: \mathbb{R}^{*} \rightarrow \mathbb{R}^{*}$ satisfying

$$f(x)+f(y)=f(x y f(x+y))$$

for $x, y \in \mathbb{R}^{*}$ and $x+y \neq 0$.