Middle European Mathematical Olympiad 2024 Problem I-3

Let $ABC$ be an acute scalene triangle. Choose a circle $\omega$ passing through $B$ and $C$ which intersects segments $AB$ and $AC$ again in points $D \neq A$ and $E \neq A$, respectively. Let $F$ be the intersection of $BE$ and $CD$. Let $G$ be the point on the circumcircle of $ABF$ such that $GB$ is tangent to $\omega$. Similarly, let $H$ be the point on the circumcircle of $ACF$ such that $HC$ is tangent to $\omega$. Prove that there exists a point $T \neq A$, independent of the choice of $\omega$, such that the circumcircle of $AGH$ passes through $T$.