
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
Let $\zeta_1, \zeta_2, \zeta_3$ be the roots of the polynomial \( x^3 - x^2 - 2x - 2 = 0 \). Knowing that \( \zeta_1 + \zeta_2 + \zeta_3 = 1\), \( \zeta_1^2 + \zeta_2^2 + \zeta_3^2 = 5 \), and \( \zeta_1^3 + \zeta_2^3 + \zeta_3^3 = 11 \), compute \( \zeta_1^5 + \zeta_2^5 + \zeta_3^5 \).

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