BALTIC-WAY 2021 Problem 13

Let $D$ be the foot of the $A$-altitude of an acute triangle $A B C$. The internal bisector of the angle $D A C$ intersects $B C$ at $K$. Let $L$ be the projection of $K$ onto $A C$. Let $M$ be the intersection point of $B L$ and $A D$. Let $P$ be the intersection point of $M C$ and $D L$. Prove that $P K \perp A B$.