Question 117
Let $ABC$ be a triangle with $AB = 13$, $AC = 14$, and $BC = 15$. Let $G$ be the point on $AC$ such that the reflection of $BG$ over the angle bisector of $\angle B$ passes through the midpoint of $AC$. Let $Y$ be the midpoint of $GC$ and $X$ be a point on segment $AG$ such that $\frac{AX}{XG} = 3$. Construct $F$ and $H$ on $AB$ and $BC$, respectively, such that $FX \parallel BG \parallel HY$. If $AH$ and $CF$ concur at $Z$ and $W$ is on $AC$ such that $WZ \parallel BG$, find $WZ$.