Hello trig, my old friend

Geometry Level 4

If the maximum of \(S=\sqrt{\sin \alpha \sin \beta \cos \gamma}+\sqrt{\sin \beta \sin \gamma \cos \alpha}+\sqrt{\sin \gamma \sin \alpha \cos \beta}\) can be represented as \(\frac{a\sqrt{b}}{c}\), where \(\alpha\), \(\beta\) and \(\gamma\) are angles of an acute-angle triangle, \(a\) and \(c\) irreducible and \(b\) square-free , compute \(a+b+c\).

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