Highly Irregular

Geometry Level 5

Each edge of a particular tetrahedron has a length of either 5 cm or 7 cm. The minimum possible volume, in $$\text{cm}^3$$, of such a tetrahedron can be written as $$\frac{a\sqrt{b}}{c}$$, where $$a$$, $$b$$, and $$c$$ are positive integers, $$a$$ and $$c$$ are coprime, and $$b$$ is square-free. Give $$a+b+c$$.

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