Školjka

Let $ABC$ be a triangle. The internal bisector of $ABC$ intersects the side $\overline{AC}$ 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 $T$. Prove $|AW| = |WT|$.