
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 \(n \geq 3\) and real numbers \(x_{1}, x_{2}, \cdots, x_{n}\) such that \(\min _{1 \leq i < j \leq n}\left|x_{i} - x_{j}\right| = 1\), find the minimum value of \(\sum_{k=1}^{n} \left|x_{k}\right|^3\).

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