
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 an integer \( n \geqslant 3 \), find the minimum value of \( \sum_{i=1}^{n}\left(\frac{1}{x_{i}}-x_{i}\right) \), where \( x_{1}, x_{2}, \cdots, x_{n} \) are positive real numbers satisfying \( \sum_{i=1}^{n} \frac{x_{i}}{x_{i}+n-1}=1 \). Additionally, find the values of \( x_i \) when the minimum value is achieved.

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