
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 $n \geq 2$, find the smallest positive real number $c$ such that for any positive real numbers $x_{1}, x_{2}, \ldots, x_{n}$, the following inequality always holds:
\[
\frac{1}{n}\left(\sum_{i=1}^{n} x_{i}\right)-\sqrt[n]{\prod_{i=1}^{n} x_{i}} \leq c \cdot \sum_{1 \leq i<j \leq n}\left(\sqrt{x_{i}}-\sqrt{x_{j}}\right)^{2}.
\]

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