
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
Simplify the products:

a) \(\sin \frac{\pi}{2 n+1} \sin \frac{2 \pi}{2 n+1} \sin \frac{3 \pi}{2 n+1} \ldots \sin \frac{n \pi}{2 n+1}\)

and

\(\sin \frac{\pi}{2 n} \sin \frac{2 \pi}{2 n} \sin \frac{3 \pi}{2 n} \ldots \sin \frac{(n-1) \pi}{2 n}\)

b) \(\cos \frac{\pi}{2 n+1} \cos \frac{2 \pi}{2 n+1} \cos \frac{3 \pi}{2 n+1} \ldots \cos \frac{n \pi}{2 n+1}\)

and

\(\cos \frac{\pi}{2 n} \cos \frac{2 \pi}{2 n} \cos \frac{3 \pi}{2 n} \ldots \cos \frac{(n-1) \pi}{2 n}\)

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