
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
Let \( x_{i} > 0 \) (for \( i = 1, 2, \cdots, n \)), \( m \in \mathbf{R}^{+} \), \( a \geq 0 \), and \( \sum_{i=1}^{n} x_{i} = s \leq n \). Then, it holds that \( \prod_{i=1}^{n} \left( x_{i}^{m} + \frac{1}{x_{i}^{m}} + a \right) \geq \left[ \left( \frac{s}{n} \right)^{m} + \left( \frac{n}{s} \right)^{m} + a \right]^{n} \).

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