2008 Fermat Problem 25

A right regular hexagonal prism is sliced as shown in the diagram. The bottom of the new solid is a regular hexagon ABCDEF. The six side faces are trapezoids perpendicular to ABCDEF. The top is a hexagon UVWXYZ that is not necessarily a regular hexagon.

Of the six edges AU, BV, CW, DX, EY, and FZ, three have lengths 4, 7 and 10. The largest possible value for AU + BV + CW + DX + EY + FZ is

\textbf{(A)}\ 42\quad \textbf{(B)}\ 51\quad \textbf{(C)}\ 69\quad \textbf{(D)}\ 78\quad \textbf{(E)}\ 91

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.