
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 \( k \geq 1 \). Find the maximum real number \( \lambda \) such that for any positive real numbers \( x_i > 0 \) \((i=1,2,\ldots,n)\) satisfying \( \sum_{i=1}^{n} \frac{1}{x_{i}} = 2016 \), the following inequality holds:
$$
\lambda \sum_{i=1}^{n} \frac{x_{i}}{1+x_{i}} \leq \left[\sum_{i=1}^{n} \frac{1}{x_{i}^{k}\left(1+x_{i}\right)}\right]\left(\sum_{i=1}^{n} x_{i}^{k}\right).
$$

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