KöMaL 2014 C1264

The interior angle bisector drawn from vertex $A$ of triangle $ABC$ intersects the opposite side at point $P$ , and the perpendicular bisector of line segment $AP$ intersects side $AC$ at point $Q$ . Given the line segments $AB$ , $AQ$ , and the angle $BAQ$ , express the area of quadrilateral $ABPQ$ .