
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 $z = \cos \frac{3 \pi}{11} + i \sin \frac{3 \pi}{11}.$ Compute the value of
\[(3z + z^3)(3z^3 + z^9)(3z^5 + z^{15})(3z^7 + z^{21})(3z^9 + z^{27})(3z^{11} + z^{33}).\]

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