HMMT 2022 G3

Let $A B C D$ and $A E F G$ be unit squares such that the area of their intersection is $\frac{20}{21}$. Given that $\angle B A E<45^{\circ}, \tan \angle B A E$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100 a+b$.