HMMT 2022 G1

Let $A B C$ be a triangle with $\angle A=60^{\circ}$. Line $\ell$ intersects segments $A B$ and $A C$ and splits triangle $A B C$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $B X$ and $C Y$ are perpendicular to $\ell$. Given that $A B=20$ and $A C=22$, compute $X Y$.