Tetrahedron $ABCD$ has the following side lengths: $AB = 5$, $AC = 3$, $BC = 4$, $BD = 4$, $AD = 3$, and $CD = \frac{12}{5}\sqrt{2}$. Find the volume of the tetrahedron. The answer is in the form \(\frac{m}{n}\), please calculate the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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