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α+β+γ=6α3+β3+γ3=87(α+1)(β+1)(γ+1)=33\large \begin{aligned}\alpha+\beta+\gamma&=6 \\\alpha^3+\beta^3+\gamma^3&=87\\ (\alpha+1)(\beta+1)(\gamma+1)&=33 \end{aligned}α+β+γα3+β3+γ3(α+1)(β+1)(γ+1)=6=87=33
Suppose α\alphaα, β\betaβ, and γ\gammaγ are complex numbers that satisfy the system of equations above.
If 1α+1β+1γ=mn\frac1\alpha+\frac1\beta+\frac1\gamma=\tfrac mnα1+β1+γ1=nm for positive coprime integers mmm and nnn, find m+nm+nm+n.
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