VTMO 2021 — Grade 9 — cup Question 5

Square $ABCD$ is inscribed in $\left(O\right)$. $M$ is the midpoint of minor arc $CD$, and $N \in AB$ with $NA = 2 NB$. $MN$ meets $\left(O\right)$ again at $E$; $F$ is $E$ reflected across $AB$. Prove $FA = 2 FB$ and $DF = 3 FB$.