Conic Sections and Parametric Equations · Challenge problem

A light ray leaves the focus $F_1=(-3,0)$ of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$, hits the ellipse at $P=(0,4)$, reflects off it, and travels on until it meets the ellipse again at a point $Q$. Find the $x$-coordinate of $Q$.