SHARYGIN 2020 Problem 15

A circle passing through the vertices $B$ and $D$ of quadrilateral $ABCD$ meets $AB$ , $BC$ , $CD$ , and $DA$ at points $K$ , $L$ , $M$ , and $N$ respectively. A circle passing through $K$ and $M$ meets $AC$ at $P$ and $Q$ . Prove that $L$ , $N$ , $P$ , and $Q$ are concyclic.