Triple System of Equations

Algebra Level 4

\[\large \begin{align*}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{align*}\]

Suppose \(\alpha\), \(\beta\), and \(\gamma\) are complex numbers that satisfy the system of equations above.

If \(\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mn\) for positive coprime integers \(m\) and \(n\), find \(m+n\).

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