JBMO-SL 2018 G1

Let $H$ be the orthocentre of an acute triangle $A B C$ with $B C>A C$, inscribed in a circle $\Gamma$. The circle with centre $C$ and radius $C B$ intersects $\Gamma$ at the point $D$, which is on the arc $A B$ not containing $C$. The circle with centre $C$ and radius $C A$ intersects the segment $C D$ at the point $K$. The line parallel to $B D$ through $K$, intersects $A B$ at point $L$. If $M$ is the midpoint of $A B$ and $N$ is the foot of the perpendicular from $H$ to $C L$, prove that the line $M N$ bisects the segment $C H$.