
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:
\[
\zeta_1 + \zeta_2 + \zeta_3 = 2
\]
\[
\zeta_1^2 + \zeta_2^2 + \zeta_3^2 = 6
\]
\[
\zeta_1^3 + \zeta_2^3 + \zeta_3^3 = 20
\]
Compute $\zeta_1^5 + \zeta_2^5 + \zeta_3^5$.

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