HMMT 2020 T8

Let $A B C$ be a scalene triangle with angle bisectors $A D, B E$, and $C F$ so that $D, E$, and $F$ lie on segments $B C, C A$, and $A B$ respectively. Let $M$ and $N$ be the midpoints of $B C$ and $E F$ respectively. Prove that line $A N$ and the line through $M$ parallel to $A D$ intersect on the circumcircle of $A B C$ if and only if $D E=D F$.