HMMT November 2024 U25

Let $A B C$ be an equilateral triangle. A regular hexagon $P X Q Y R Z$ of side length 2 is placed so that $P, Q$, and $R$ lie on segments $\overline{B C}, \overline{C A}$, and $\overline{A B}$, respectively. If points $A, X$, and $Y$ are collinear, compute $B C$.