
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 \(\zeta_1, \zeta_2,\) and \(\zeta_3\) be complex numbers such that:
\[\zeta_1+\zeta_2+\zeta_3=2,\]
\[\zeta_1^2+\zeta_2^2+\zeta_3^2=5,\]
\[\zeta_1^3+\zeta_2^3+\zeta_3^3=8.\]
Compute \(\zeta_1^8 + \zeta_2^8 + \zeta_3^8\).

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