
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
Find all natural values of \( n \) for which \( \cos \frac{2 \pi}{9} + \cos \frac{4 \pi}{9} + \cdots + \cos \frac{2 \pi n}{9} = \cos \frac{\pi}{9} \), and \( \log_{3}^{2} n + 14 < \log_{3} 9 n^{7} \).

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