
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
(1) Let \( n \) be a prime number greater than 3. Find the value of \(\left(1 + 2 \cos \frac{2\pi}{n}\right) \left(1 + 2 \cos \frac{4\pi}{n}\right) \left(1 + 2 \cos \frac{6\pi}{n}\right) \cdots \left(1 + 2 \cos \frac{2n\pi}{n}\right).

(2) Let \( n \) be a natural number greater than 3. Find the value of \(\left(1 + 2 \cos \frac{\pi}{n}\right) \left(1 + 2 \cos \frac{2\pi}{n}\right) \left(1 + 2 \cos \frac{3\pi}{n}\right) \cdots \left(1 + 2 \cos \frac{(n-1)\pi}{n}\right).

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