OIM 2017 Problem 2

Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$ , different from $B$ and $C$ , and let $M$ be the midpoint of $AD$ . The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$ , with point $D$ between $E$ and $F$ . Lines $FC$ and $EM$ intersect at point $X$ . If $\angle DAE = \angle AFE$ , show that line $AX$ is tangent to $\Gamma$ .