
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:

$$\left( \sin\frac{\pi}{3} \right)^{-2}+\left( \sin\frac{2\pi}{3} \right)^{-2}=\frac{4}{3}\times1\times2;$$

$$\left( \sin\frac{\pi}{5} \right)^{-2}+\left( \sin\frac{2\pi}{5} \right)^{-2}+\left( \sin\frac{3\pi}{5} \right)^{-2}+\left( \sin\frac{4\pi}{5} \right)^{-2}=\frac{4}{3}\times2\times3;$$

$$\left( \sin\frac{\pi}{7} \right)^{-2}+\left( \sin\frac{2\pi}{7} \right)^{-2}+\left( \sin\frac{3\pi}{7} \right)^{-2}+\cdots+\left( \sin\frac{6\pi}{7} \right)^{-2}=\frac{4}{3}\times3\times4;$$

$$\left( \sin\frac{\pi}{9} \right)^{-2}+\left( \sin\frac{2\pi}{9} \right)^{-2}+\left( \sin\frac{3\pi}{9} \right)^{-2}+\cdots+\left( \sin\frac{8\pi}{9} \right)^{-2}=\frac{4}{3}\times4\times5;$$

$$\vdots$$

According to this pattern, $\left( \sin\frac{\pi}{2n+1} \right)^{-2}+\left( \sin\frac{2\pi}{2n+1} \right)^{-2}+\left( \sin\frac{3\pi}{2n+1} \right)^{-2}+\cdots+\left( \sin\frac{2n\pi}{2n+1} \right)^{-2}=$\_\_\_\_$.

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