Linear Transformations · Superchallenge problem

Let $P$ be the orthogonal projection of $\mathbb{R}^2$ onto the line $y = 2x$ and $F$ the reflection across the line $y = x$. Let $G = P\circ F$ (apply $F$ first). Find the $x$-coordinate of $G^5(5, 0)$, the result of applying $G$ five times to $(5,0)$.