HMMT November 2024 U18

Let $A B C D$ be a rectangle whose vertices are labeled in counterclockwise order with $A B=32$ and $A D=60$. Rectangle $A B^{\prime} C^{\prime} D^{\prime}$ is constructed by rotating $A B C D$ counterclockwise about $A$ by $60^{\circ}$. Given that lines $B B^{\prime}$ and $D D^{\prime}$ intersect at point $X$, compute $C X$.