HMMT 2021 U18

Triangle $A B C$ has side lengths $A B=19, B C=20$, and $C A=21$. Points $X$ and $Y$ are selected on sides $A B$ and $A C$, respectively, such that $A Y=X Y$ and $X Y$ is tangent to the incircle of $\triangle A B C$. If the length of segment $A X$ can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, compute $100 a+b$.