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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