
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 \(\left(1+2 \cos \frac{2 \pi}{n}\right)\left(1+2 \cos \frac{4 \pi}{n}\right) \cdots\left(1+2 \cos \frac{2 m \pi}{n}\right)\).

(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}
user