Let $\mathbb{Z}^{+}$ be the set of all positive integers. Find all functions $f: \mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ satisfying the following conditions for all $x, y \in \mathbb{Z}^{+}$:

$$
\begin{aligned}
f(x, x) & =x, \\
f(x, y) & =f(y, x), \\
(x+y) f(x, y) & =y f(x, x+y) .
\end{aligned}
$$