PAGMO 2022 Problem 3

Let $ABC$ be an acute triangle with $AB< AC$ . Denote by $P$ and $Q$ points on the segment $BC$ such that $\angle BAP = \angle CAQ < \frac{\angle BAC}{2}$ . $B_1$ is a point on segment $AC$ . $BB_1$ intersects $AP$ and $AQ$ at $P_1$ and $Q_1$ , respectively. The angle bisectors of $\angle BAC$ and $\angle CBB_1$ intersect at $M$ . If $PQ_1\perp AC$ and $QP_1\perp AB$ , prove that $AQ_1MPB$ is cyclic.