Not Newton's Sum

Algebra Level 4

Consider the cubic equation \( 245x^3 - 287x^2 + 99x - 9 = 0 \) with roots \( \alpha , \beta , \gamma \). If

\[ \displaystyle\sum_{r=0}^{\infty} \left ( \alpha^{r} + \beta^{r} + \gamma^{r} \right ) \]

is of the form \( \frac {m}{n} \), where \(m\) and \(n\) are coprime positive integers, what is value of \( \frac {m}{n+1} \)?

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