Given an acute-angled triangle $ABC$ ($AB \neq AC$) with orthocenter $H$, circumcenter $O$, and midpoint $M$ of side $BC$. The line $AM$ intersects the circumcircle of triangle $BHC$ at point $K$, with $M$ between $A$ and $K$. The segments $HK$ and $BC$ intersect at point $N$. If $\angle BAM = \angle CAN$, prove that the lines $AN$ and $OH$ are perpendicular.