
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 \( m \geq 2 \), find the maximum real number \( C \) such that the inequality 
\[ \left( \sum_{k=1}^{m} \frac{1}{x_{k}} \right) \left( \sum_{k=1}^{m} (x_{k}^3 + 2x_{k}) \right) - \left( \sum_{k=1}^{m} \sqrt{x_{k}^2 + 1} \right)^2 \geq C \]
holds for any \( m \) distinct positive integers \( x_{1}, x_{2}, \ldots, x_{m} \).

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