US-TST 2019 Problem 1

Let $A B C$ be a triangle and let $M$ and $N$ denote the midpoints of $\overline{A B}$ and $\overline{A C}$, respectively. Let $X$ be a point such that $\overline{A X}$ is tangent to the circumcircle of triangle $A B C$. Denote by $\omega_{B}$ the circle through $M$ and $B$ tangent to $\overline{M X}$, and by $\omega_{C}$ the circle through $N$ and $C$ tangent to $\overline{N X}$. Show that $\omega_{B}$ and $\omega_{C}$ intersect on line $B C$.