
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
Given complex numbers \( \zeta_1, \zeta_2, \) and \( \zeta_3 \) such that
\[ \zeta_1 + \zeta_2 + \zeta_3 = 1, \]
\[ \zeta_1^2 + \zeta_2^2 + \zeta_3^2 = 5, \]
\[ \zeta_1^3 + \zeta_2^3 + \zeta_3^3 = 9, \]
compute \( \zeta_1^6 + \zeta_2^6 + \zeta_3^6 \).

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