
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
Using the result of problem 23.6, compute the following sums and products:

a) \(\operatorname{ctg}^{2} \frac{\pi}{2n+1} + \operatorname{ctg}^{2} \frac{2\pi}{2n+1} + \ldots + \operatorname{ctg}^{2} \frac{n\pi}{2n+1}\);

b) \(\frac{1}{\sin^{2} \frac{\pi}{2n+1}} + \frac{1}{\sin^{2} \frac{2\pi}{2n+1}} + \ldots + \frac{1}{\sin^{2} \frac{n\pi}{2n+1}}\);

c) \(\sin \frac{\pi}{2n+1} \sin \frac{2\pi}{2n+1} \ldots \sin \frac{n\pi}{2n+1}\).

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