OIM 2016 Problem 3

Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$ . Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$ , and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$ . Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$ . Prove that $L, J, A, Q,$ and $K$ all lie on a circle.