
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  $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$  and let  $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$ . Find the value of the following expression:   $$ f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4). $$  

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