In regular hexagon $ABCDEF$, points $W,X,Y,Z$ lie on sides $\overline{BC},\overline{CD},\overline{EF},\overline{FA}$, respectively. Lines $AB$, $ZW$, $YX$, and $ED$ are parallel, with the distance from $AB$ to $ZW$ equal to the distance from $YX$ to $ED$, and each of these distances twice the distance from $ZW$ to $YX$. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$?

1. Let $h$ be the distance from the center of the regular hexagon to side $AB$, so the distance from $AB$ to $ED$ is $2h$. The three gaps have ratio $2:1:2$, so the lower and upper gaps each have height $\frac{4h}{5}$.
2. Since the hexagon has side length $1$, the parallel cross-sections $ZW$ and $YX$ each have length $\frac95$. Each corner trapezoid, such as $ABWZ$, has area $\frac{1+\frac95}{2}\cdot\frac{4h}{5}=\frac{28h}{25}$.
3. The whole hexagon has area $3h$. Therefore $[WCXYFZ]=3h-2\cdot\frac{28h}{25}=\frac{19h}{25}$, and $\dfrac{[WCXYFZ]}{[ABCDEF]}=\boxed{\dfrac{19}{75}}$.