
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 \( x_{i} \geq 0 \) for \( i = 1, 2, \cdots, n \) and \( \sum_{i=1}^{n} x_{i} = 1 \) with \( n \geq 2 \), find the maximum value of \( \sum_{1 \leq i \leq j \leq n} x_{i} x_{j} (x_{i} + x_{j}) \).

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