SHARYGIN 2022 Problem 82

Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$ . Circumcircles $\omega_{1}$ and $\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$ . Let $ED$ meet $\omega_{1}$ at point $F$ , and $FB$ meet $AD$ at point $G$ . Prove that $GM$ bisects angle $BGD$ .