
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 \( n \geq 3 \), if there exists a positive real number \( k > 1 \) such that for any positive real numbers \( x_{1}, x_{2}, \cdots x_{n} \) satisfying \( \sum_{i=1}^{n} x_{i} = k \sum_{i=1}^{n} \frac{1}{x_{i}} \), the inequality \( \sum_{i=1}^{n} \frac{x_{i}^{2}}{x_{i}^{2}+1} \leq k \sum_{i=1}^{n} \frac{1}{x_{i}^{2}+1} \) must hold, then \( k \) is called a "good number". Find the smallest good number \( k \).

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