CONO-SUR 2020 Problem 3

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$ . Points $F$ and $E$ are chosen in $AB$ and $BC$ , respectively, such that $AC=CF$ and $EB=EF$ . The line $AM$ intersects $\omega$ in $D\neq A$ . The line $DE$ intersects the line $FM$ in $G$ . Prove that $G$ lies on $\omega$ .