Tetrahedral Volume Sum

Geometry Level 5

Consider a tetrahedron with side lengths 2,3,3,4,5,52, 3, 3, 4, 5, 5. The largest possible volume of this tetrahedron has the form abc \frac {a \sqrt{b}}{c}, where bb is an integer that's not divisible by the square of any prime, aa and cc are positive, coprime integers. What is the value of a+b+ca+b+c?

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