
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 \( n > 1 \) be an odd number. Find the value of \( \prod_{k=1}^{m} \cos \frac{a_{k} \pi}{n} \), where \( a_{1}, a_{2}, \cdots, a_{m} \) are all positive integers less than or equal to \( n \) that are coprime to \( n \).

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