
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
Compose equations whose roots are the numbers:

a) \(\operatorname{ctg}^{2} \frac{\pi}{2 m+1}, \operatorname{ctg}^{2} \frac{2 \pi}{2 m+1}, \operatorname{ctg}^{2} \frac{3 \pi}{2 m+1}, \ldots, \operatorname{ctg}^{2} \frac{(m-1) \pi}{2 m+1}, \operatorname{ctg}^{2} \frac{m \pi}{2 m+1}\)

b) \(\operatorname{ctg} \frac{\pi}{4 n}, -\operatorname{ctg} \frac{3 \pi}{4 n}, \operatorname{ctg} \frac{5 \pi}{4 n}, -\operatorname{ctg} \frac{7 \pi}{4 n}, \ldots, \operatorname{ctg} \frac{(2 n-3) \pi}{4 n}, -\operatorname{ctg} \frac{(2 n-1) \pi}{4 n}\) (for \(n\) even)

c) \(\sin^{2} \frac{\pi}{2 m}, \sin^{2} \frac{2 \pi}{2 m}, \sin^{2} \frac{3 \pi}{2 m}, \ldots, \sin^{2} \frac{(m-1) \pi}{2 m}\);

d) \(\sin^{2} \frac{\pi}{4 m}, \sin^{2} \frac{3 \pi}{4 m}, \sin^{2} \frac{5 \pi}{4 m}, \ldots, \sin^{2} \frac{(2 m-1) \pi}{4 m}\).

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