SHARYGIN 2022 Problem 9

The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$ . Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$ , and $RM$ meet $DI$ at point $S$ . Prove that $P,N$ and $S$ are collinear.