
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 \( a_1, a_2, \ldots, a_n \) that satisfy \( \min_{1 \leq i < j \leq n} \left|a_i - a_j\right| = 1 \), find the minimum value of \( \sum_{k=1}^n \left| a_k \right|^3 \).

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