
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 number $\lambda$ and a positive integer $n$ (where $n \geqslant 2$), find the smallest positive number $M(\lambda)$ such that for all nonnegative real numbers $x_{1}, x_{2}, \cdots, x_{n}$, the following inequality holds:
\[ M(\lambda) \left( \sum_{k=1}^{n} x_{k} \right)^{n} \geqslant \sum_{k=1}^{n} x_{k}^{n} + \lambda \prod_{k=1}^{n} x_{k}. \]

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