SHARYGIN 2019 Problem 14

Let the side $AC$ of triangle $ABC$ touch the incircle and the corresponding excircle at points $K$ and $L$ respectively. Let $P$ be the projection of the incenter onto the perpendicular bisector of $AC$ . It is known that the tangents to the circumcircle of triangle $BKL$ at $K$ and $L$ meet on the circumcircle of $ABC$ . Prove that the lines $AB$ and $BC$ touch the circumcircle of triangle $PKL$ .