
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
Determine the largest real number \( C \) such that the inequality
\[
\sum_{k=1}^{100} \frac{1}{x_{k}} \sum_{k=1}^{100}\left(x_{k}^{3}+2 x_{k}\right)-\left(\sum_{k=1}^{100} \sqrt{x_{k}^{2}+1}\right)^{2} \geqslant C
\]
holds for any 100 distinct positive integers \( x_{1}, x_{2}, \ldots, x_{100} \).

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