Middle European Mathematical Olympiad 2018 Problem T-6

Let $ABC$ be a triangle. The internal bisector of $\angle ABC$ intersects the side $AC$ at $L$ and the circumcircle of triangle $ABC$ again at $W \neq B$. Let $K$ be the perpendicular projection of $L$ onto $AW$. The circumcircle of triangle $BLC$ intersects line $CK$ again at $P \neq C$. Lines $BP$ and $AW$ meet at point $T$. Prove that $AW = WT$.