CWMO 2011 Problem 4

In a circle $\Gamma_{1}$ , centered at $O$ , $AB$ and $CD$ are two unequal in length chords intersecting at $E$ inside $\Gamma_{1}$ . A circle $\Gamma_{2}$ , centered at $I$ is tangent to $\Gamma_{1}$ internally at $F$ , and also tangent to $AB$ at $G$ and $CD$ at $H$ . A line $l$ through $O$ intersects $AB$ and $CD$ at $P$ and $Q$ respectively such that $EP = EQ$ . The line $EF$ intersects $l$ at $M$ . Prove that the line through $M$ parallel to $AB$ is tangent to $\Gamma_{1}$