HMMT 2021 G4

Let $A B C D$ be a trapezoid with $A B \| C D, A B=5, B C=9, C D=10$, and $D A=7$. Lines $B C$ and $D A$ intersect at point $E$. Let $M$ be the midpoint of $C D$, and let $N$ be the intersection of the circumcircles of $\triangle B M C$ and $\triangle D M A$ (other than $M$ ). If $E N^{2}=\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100 a+b$.