
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
Consider a monic polynomial of degree $n$ with real coefficients given by $x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + a_{n-3}x^{n-3} + \cdots + a_0$. It is known that $a_{n-1} = -a_{n-2}$ and $a_{n-3} = 2a_{n-2}$. Determine the greatest lower bound of the absolute value of the sum of the cubes of the roots of the polynomial.

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