INMO 2010 Problem 1

Let $A B C$ be a triangle with circum-circle $\Gamma$. Let $M$ be a point in the interior of triangle $A B C$ which is also on the bisector of $\angle A$. Let $A M, B M, C M$ meet $\Gamma$ in $A_{1}, B_{1}, C_{1}$ respectively. Suppose $P$ is the point of intersection of $A_{1} C_{1}$ with $A B$; and $Q$ is the point of intersection of $A_{1} B_{1}$ with $A C$. Prove that $P Q$ is parallel to $B C$.