Tetrahedral Volume Sum

Geometry Level 5

Consider a tetrahedron with side lengths $$2, 3, 3, 4, 5, 5$$. The largest possible volume of this tetrahedron has the form $$\frac {a \sqrt{b}}{c}$$, where $$b$$ is an integer that's not divisible by the square of any prime, $$a$$ and $$c$$ are positive, coprime integers. What is the value of $$a+b+c$$?

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