SMO 2022 Problem 1

For $\triangle ABC$ and its circumcircle $\omega$, draw the tangents at $B,C$ to $\omega$ meeting at $D$. Let the line $AD$ meet the circle with center $D$ and radius $DB$ at $E$ inside $\triangle ABC$. Let $F$ be the point on the extension of $EB$ and $G$ be the point on the segment $EC$ such that $\angle AFB=\angle AGE=\angle A$. Prove that the tangent at $A$ to the circumcircle of $\triangle AFG$ is parallel to $BC$.

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