
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 complex numbers such that:
\[\zeta_1+\zeta_2+\zeta_3 = 2,\]
\[\zeta_1^2+\zeta_2^2+\zeta_3^2 = 6,\]
\[\zeta_1^3+\zeta_2^3+\zeta_3^3 = 8.\]
Compute $\zeta_1^7 + \zeta_2^7 + \zeta_3^7$.

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