SHARYGIN 2010 Problem 6

Points $M$ and $N$ lie on the side $BC$ of a regular triangle $ABC$ ($M$ is between $B$ and $N$), and $\angle MAN = 30^\circ$. The circumcircles of the triangles $AMC$ and $ANB$ meet at a point $K$. Prove that the line $AK$ passes through the circumcenter of the triangle $AMN$.