Not just AM-GM

Algebra Level 4

\[\frac{1}{a^{3}(b+c)} + \frac{1}{b^{3}(c+a)} + \frac{1}{c^{3}(a+b)} \geq \frac{A}{B}\]

The inequality above, where \(A\) and \(B\) are coprime, holds true for \(abc = 1\) and \(a, b, c > 0\). Find \(A+B\).

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