Hari is obsessed with cubics. He comes up with a cubic polynomial with leading coefficient 1, rational coefficients, and real roots $0 < a < b < c < 1$. He knows the following three facts:  
1. $P(0) = -\frac{1}{8}$
2. The roots form a geometric progression in the order $a, b, c$.
3. \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \frac{9}{2} \]

The value $a + b + c$ can be expressed as $\frac{m}{n}$, where $m, n$ are relatively prime positive integers. Find $m + n$.

Let's think step by step and output the final answer within \boxed{}.
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