
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 \( n \in \mathbf{N}, n \geq 2 \), and non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) such that \( x_{1} + x_{2} + \cdots + x_{n} = 1 \). Find the maximum and minimum values of \( \sum_{1 \leq i < j \leq n} x_{i} x_{j} \left(x_{i} + x_{j} \right) \), and determine when these values are attained.

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