FKMO 2018 Problem 4

Triangle $ABC$ satisfies $\angle C=90^{\circ}$ . A circle passing $A,B$ meets segment $AC$ at $G(\neq A,C)$ and it meets segment $BC$ at point $D(\neq B)$ . Segment $AD$ cuts segment $BG$ at $H$ , and let $l$ , the perpendicular bisector of segment $AD$ , cuts the perpendicular bisector of segment $AB$ at point $E$ . A line passing $D$ is perpendicular to $DE$ and cuts $l$ at point $F$ . If the circumcircle of triangle $CFH$ cuts $AC$ , $BC$ at $P(\neq C),Q(\neq C)$ respectively, then prove that $PQ$ is perpendicular to $FH$ .