BALKAN-SL 2021 G1

Let $A B C$ be a triangle with $A B<A C<B C$. On the side $B C$ we consider points $D$ and $E$ such that $B A=B D$ and $C E=C A$. Let $K$ be the circumcenter of triangle $A D E$ and let $F, G$ be the points of intersection of the lines $A D, K C$ and $A E, K B$ respectively. Let $\omega_{1}$ be the circumcircle of triangle $K D E, \omega_{2}$ the circle with center $F$ and radius $F E$, and $c_{3}$ the circle with center $G$ and radius $G D$.

Prove that $\omega_{1}, \omega_{2}$ and $\omega_{3}$ pass through the same point and that this point of intersection lies on the line $A K$.

Proposed by Greece