FKMO 2013 Problem 1

For a triangle $\triangle ABC (\angle B > \angle C)$ , $D$ is a point on $AC$ satisfying $\angle ABD = \angle C$ . Let $I$ be the incenter of $\triangle ABC$ , and circumcircle of $\triangle CDI$ meets $AI$ at $E ( \ne I )$ . The line passing $E$ and parallel to $AB$ meets the line $BD$ at $P$ . Let $J$ be the incenter of $\triangle ABD$ , and $A'$ be the point such that $AI = IA'$ . Let $Q$ be the intersection point of $JP$ and $A'C$ . Prove that $QJ = QA'$ .