MK-MO 2009 Problem 2

Let $O$ be the centre of the incircle of $\triangle ABC$ . Points $K,L$ are the intersection points of the circles circumscribed about triangles $BOC,AOC$ respectively with the bisectors of the angles at $A,B$ respectively $(K,L\not= O)$ . Also $P$ is the midpoint of segment $KL$ , $M$ is the reflection of $O$ with respect to $P$ and $N$ is the reflection of $O$ with respect to line $KL$ . Prove that the points $K,L,M$ and $N$ lie on the same circle.