
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 $w = \cos \frac{6 \pi}{11} + i \sin \frac{6 \pi}{11}.$ Compute
\[
\frac{w}{1 + w^2} + \frac{w^3}{1 + w^6}+\frac{w^4}{1 + w^8}.
\]

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