
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
Observe the following equations:

$(\sin \frac{\pi}{3})^{-2} + (\sin \frac{2\pi}{3})^{-2} = \frac{4}{3} \times 1 \times 2$;

$(\sin \frac{\pi}{5})^{-2} + (\sin \frac{2\pi}{5})^{-2} + (\sin \frac{3\pi}{5})^{-2} + (\sin \frac{4\pi}{5})^{-2} = \frac{4}{3} \times 2 \times 3$;

$(\sin \frac{\pi}{7})^{-2} + (\sin \frac{2\pi}{7})^{-2} + (\sin \frac{3\pi}{7})^{-2} + \ldots + (\sin \frac{6\pi}{7})^{-2} = \frac{4}{3} \times 3 \times 4$;

$(\sin \frac{\pi}{9})^{-2} + (\sin \frac{2\pi}{9})^{-2} + (\sin \frac{3\pi}{9})^{-2} + \ldots + (\sin \frac{8\pi}{9})^{-2} = \frac{4}{3} \times 4 \times 5$;

$...$

Following this pattern, $(\sin \frac{\pi}{2n+1})^{-2} + (\sin \frac{2\pi}{2n+1})^{-2} + (\sin \frac{3\pi}{2n+1})^{-2} + \ldots + (\sin \frac{2n\pi}{2n+1})^{-2} = \_\_\_\_\_\_\_\_\_\_$.

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