APMO 2018 Problem 1

Let $H$ be the orthocenter of the triangle $A B C$. Let $M$ and $N$ be the midpoints of the sides $A B$ and $A C$, respectively. Assume that $H$ lies inside the quadrilateral $B M N C$ and that the circumcircles of triangles $B M H$ and $C N H$ are tangent to each other. The line through $H$ parallel to $B C$ intersects the circumcircles of the triangles $B M H$ and $C N H$ in the points $K$ and $L$, respectively. Let $F$ be the intersection point of $M K$ and $N L$ and let $J$ be the incenter of triangle $M H N$. Prove that $F J=F A$.