SHARYGIN 2023 Problem 1

Let $L$ be the midpoint of the minor arc $AC$ of the circumcircle of an acute-angled triangle $ABC$ . A point $P$ is the projection of $B$ to the tangent at $L$ to the circumcircle. Prove that $P$ , $L$ , and the midpoints of sides $AB$ , $BC$ are concyclic.