BALKAN-SL 2018 G5

Let $A B C$ be an acute-angled triangle with $A B<A C<B C$ and let $D$ be an arbitrary point on the extension of $B C$ beyond $C$. The circle $\gamma(A, A D)$ intersects the rays $A C$, $A B, C B$ at points $E, F, G$, respectively. The circumcircle $\omega_{1}$ of triangle $A F G$ intersects the lines $F E, B C, G E, D F$ again at points $J, H, H^{\prime}, J^{\prime}$. The circumcircle $\omega_{2}$ of triangle $A D E$ intersects the lines $F E, B C, G E, D F$ again at points $I, K, K^{\prime}, I^{\prime}$. Prove that the quadrilaterals $H I J K$ and $H^{\prime} I^{\prime} J^{\prime} K^{\prime}$ are cyclic and that their circumcenters coincide. (Greece)