CAPS 2014 Problem 3

Given is a convex $ABCD$ , which is $|\angle ABC| = |\angle ADC|= 135^\circ$ . On the $AB, AD$ are also selected points $M, N$ such that $|\angle MCD| = |\angle NCB| = 90^ \circ$ . The circumcircles of the triangles $AMN$ and $ABD$ intersect for the second time at point $K \ne A$ . Prove that lines $AK$ and $KC$ are perpendicular.

(Irán)