In equilateral triangle $XYZ$ with side length $10$, define points $A, B$ on $XY,$ points $C, D$ on $YZ,$ and points $E, F$ on $ZX$ such that $ABDE$ and $ACEF$ are rectangles, $XA<XB,$ $YC<YD,$ and $ZE<ZF$. The area of hexagon $ABCDEF$ can be written as $\sqrt{x}$ for some positive integer $x$. Find $x$.