SHARYGIN 2011 Problem 21

On a circle with diameter $AC$ , let $B$ be an arbitrary point distinct from $A$ and $C$ . Points $M, N$ are the midpoints of chords $AB, BC$ , and points $P, Q$ are the midpoints of smaller arcs restricted by these chords. Lines $AQ$ and $BC$ meet at point $K$ , and lines $CP$ and $AB$ meet at point $L$ . Prove that lines $MQ, NP$ and $KL$ concur.