AMC 12 2015a Problem 16

Tetrahedron $ABCD$ has $AB=5,AC=3,BC=4,BD=4,AD=3,$ and $CD=\frac{12}{5}\sqrt{2}$. What is the volume of the tetrahedron?

\textbf{(A)}\ 3\sqrt{2} \textbf{(B)}\ 2\sqrt{5} \textbf{(C)}\ \frac{24}{5} \textbf{(D)}\ 3\sqrt{3} \textbf{(E)}\ \frac{24}{5}\sqrt{2}