HMMT 2016 U20

Let $A B C$ be a triangle with $A B=13, A C=14$, and $B C=15$. Let $G$ be the point on $A C$ such that the reflection of $B G$ over the angle bisector of $\angle B$ passes through the midpoint of $A C$. Let $Y$ be the midpoint of $G C$ and $X$ be a point on segment $A G$ such that $\frac{A X}{X G}=3$. Construct $F$ and $H$ on $A B$ and $B C$, respectively, such that $F X\|B G\| H Y$. If $A H$ and $C F$ concur at $Z$ and $W$ is on $A C$ such that $W Z \| B G$, find $W Z$.