
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 \( n \) be a positive integer. Find the value of the product \(\left(1 + 2 \cos \frac{2 \pi}{2n+1}\right) \left(1 + 2 \cos \frac{4 \pi}{2n+1}\right) \cdots \left(1 + 2 \cos \frac{4n \pi}{2n+1}\right) \).

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