HMMT 2016 G6

Let $A B C$ be a triangle with incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M, N, P$ be the midpoints of sides $\overline{B C}, \overline{C A}, \overline{A B}$ and let $E, F$ be the tangency points of $\gamma$ with $\overline{C A}$ and $\overline{A B}$, respectively. Let $U$, $V$ be the intersections of line $E F$ with line $M N$ and line $M P$, respectively, and let $X$ be the midpoint of arc $\widehat{B A C}$ of $\Gamma$. Given that $A B=5, A C=8$, and $\angle A=60^{\circ}$, compute the area of triangle $X U V$.