
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
Given a positive integer \( n \geq 2 \) and \( x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}^{+} \) such that \( x_{1}+x_{2}+\cdots+x_{n}=\pi \), find the minimum value of \( \left(\sin x_{1}+\frac{1}{\sin x_{1}}\right)\left(\sin x_{2}+\frac{1}{\sin x_{2}}\right) \cdots\left(\sin x_{n}+\frac{1}{\sin x_{n}}\right) \).

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