HMMT November 2020 U29

In acute triangle $A B C$, let $H$ be the orthocenter and $D$ the foot of the altitude from $A$. The circumcircle of triangle $B H C$ intersects $A C$ at $E \neq C$, and $A B$ at $F \neq B$. If $B D=3, C D=7$, and $\frac{A H}{H D}=\frac{5}{7}$, the area of triangle $A E F$ can be expressed as $\frac{a}{b}$, where $a, b$ are relatively prime positive integers. Compute $100 a+b$.