What is the volume of tetrahedron $ABCD$ with edge lengths $AB = 5$, $AC = 6$, $AD = 7$, $BC = \sqrt{61}$, $BD = \sqrt{74}$, and $CD = \sqrt{85}$ ? $\textbf{(A)} ~32 \qquad\textbf{(B)} ~35\sqrt{3} \qquad\textbf{(C)} ~42\qquad\textbf{(D)} ~56\sqrt{3}\qquad\textbf{(E)} ~63$