
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Hari is obsessed with cubics.  He comes up with a cubic with leading coefficient 1, rational coefficients and real roots  $0 < a < b < c < 1$ .  He knows the following three facts:  $P(0) = -\frac{1}{8}$ , the roots form a geometric progression in the order  $a,b,c$ , and \[ \sum_{k=1}^{\infty} (a^k + b^k + c^k) = \dfrac{9}{2} \]  The value  $a + b + c$  can be expressed as  $\frac{m}{n}$ , where  $m,n$  are relatively prime positive integers.  Find  $m + n$ .

*Proposed by Akshar Yeccherla (TopNotchMath)*

Present the answer in LaTex format: \boxed{Your answer}
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