CN-TST 2018 Problem 21

In isosceles $\triangle ABC$ , $AB=AC$ , points $D,E,F$ lie on segments $BC,AC,AB$ such that $DE\parallel AB$ , $DF\parallel AC$ . The circumcircle of $\triangle ABC$ $\omega_1$ and the circumcircle of $\triangle AEF$ $\omega_2$ intersect at $A,G$ . Let $DE$ meet $\omega_2$ at $K\neq E$ . Points $L,M$ lie on $\omega_1,\omega_2$ respectively such that $LG\perp KG, MG\perp CG$ . Let $P,Q$ be the circumcenters of $\triangle DGL$ and $\triangle DGM$ respectively. Prove that $A,G,P,Q$ are concyclic.