RMO 2014 B5

Let $ABC$ be a triangle with $AB > AC$ . Let $P$ be a point on the line $AB$ beyond $A$ such that $AP +P C = AB$ . Let $M$ be the mid-point of $BC$ and let $Q$ be the point on the side $AB$ such that $CQ \perp AM$ . Prove that $BQ = 2AP.$