
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 natural number \( n \geq 3 \) and real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) satisfying \( x_{1} + x_{2} + \cdots + x_{n} = n \) and \( x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2} = n^{2} \), find the minimum value of \( \sum_{i=1}^{n} x_{i}^{3} \) and the corresponding values of \( \left(x_{1}, x_{2}, \cdots, x_{n}\right) \) that achieve this minimum.

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