Olympiad Geometry · Superchallenge problem

A right prism has regular hexagons $ABCDEF$ (bottom) and $A'B'C'D'E'F'$ (top) as its bases, with side length $2$, and height $6$ ($A'$ is directly above $A$, and so on). A plane contains the bottom edge $\overline{AB}$ and the top edge $\overline{D'E'}$. Find the area of the cross-section of the prism cut by this plane, in simplest radical form.