BxMO 2012 Problem 3

In triangle $A B C$ the midpoint of $B C$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle C P M=\angle P A B$. Let $\Gamma$ be the circumcircle of triangle $A B P$. The line $M P$ intersects $\Gamma$ a second time in $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ in $B$. Prove that the length $|Q R|$ is independent of the position of $P$ inside the triangle.