TH-TMO 2018 Problem 9

In $\vartriangle ABC$ the incircle is tangent to $AB$ at $D$ . Let $P$ be a point on $BC$ different from $B$ and $C$ , and let $K$ and $L$ be incenters of $\vartriangle ABP$ and $\vartriangle ACP$ respectively. Suppose that the circumcircle of $\vartriangle KP L$ cuts $AP$ again at $Q$ . Prove that $AD = AQ$ .