CR-MO 2017 G4

In triangle $ABC$ with incenter $I$ and circumcircle $\omega$ , the tangent through $C$ to $\omega$ intersects $AB$ at point $D$ . The angle bisector of $\angle CDB$ intersects $AI$ and $BI$ at $E$ and $F$ , respectively. Let $M$ be the midpoint of $[EF]$ . Prove that line $MI$ passes through the midpoint of arc $ACB$ of $w$ .