MEMO 2014 Problem 6

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$ . Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$ . Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.