KöMaL problem A, 2020

Let $\omega$ denote the incircle of triangle $ABC$, which is tangent to side $BC$ at point $D$. Let $G$ denote the second intersection of line $AD$ and circle $\omega$. The tangent to $\omega$ at point $G$ intersects sides $AB$ and $AC$ at points $E$ and $F$. The circumscribed circle of $DEF$ intersects $\omega$ at points $D$ and $M$. The circumscribed circle of $BCG$ intersects $\omega$ at point $G$ and $N$. Prove that lines $AD$ and $MN$ are parallel.