Triple System of Equations

Algebra Level 3

$\large \begin{aligned}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{aligned}$

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 $m+n$ .