KMO 2020 Problem 6

Let $ABCDE$ be a convex pentagon such that quadrilateral $ABDE$ is a parallelogram and quadrilateral $BCDE$ is inscribed in a circle. The circle with center $C$ and radius $CD$ intersects the line $BD, DE$ at points $F, G(\neq D)$ , and points $A, F, G$ is on line l. Let $H$ be the intersection point of line $l$ and segment $BC$ . Consider the set of circle $\Omega$ satisfying the following condition.

Circle $\Omega$ passes through $A, H$ and intersects the sides $AB, AE$ at point other than $A$ .

Let $P, Q(\neq A)$ be the intersection point of circle $\Omega$ and sides $AB, AE$ . Prove that $AP+AQ$ is constant.