HMMT 2020 G2

Let $A B C$ be a triangle with $A B=5, A C=8$, and $\angle B A C=60^{\circ}$. Let $U V W X Y Z$ be a regular hexagon that is inscribed inside $A B C$ such that $U$ and $V$ lie on side $B A, W$ and $X$ lie on side $A C$, and $Z$ lies on side $C B$. What is the side length of hexagon $U V W X Y Z$ ?