KMO 2021 Problem 6

Let $ABC$ be an obtuse triangle with $\angle A > \angle B > \angle C$ , and let $M$ be a midpoint of the side $BC$ . Let $D$ be a point on the arc $AB$ of the circumcircle of triangle $ABC$ not containing $C$ . Suppose that the circle tangent to $BD$ at $D$ and passing through $A$ meets the circumcircle of triangle $ABM$ again at $E$ and $\overline{BD}=\overline{BE}$ . $\omega$ , the circumcircle of triangle $ADE$ , meets $EM$ again at $F$ .

Prove that lines $BD$ and $AE$ meet on the line tangent to $\omega$ at $F$ .