CWMO 2019 Problem 2

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$ with $AB\neq AC$ , respectively. Let $M$ be the midpoint of $BC$ and $K$ be the intersection of $AM$ and the circumcircle of $\triangle BHC$ , such that $M$ lies between $A$ and $K$ . Let $N$ be the intersection of $HK$ and $BC$ . Show that if $\angle BAM=\angle CAN$ , then $AN\perp OH$ .