
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 \( n \geq 2 \), determine the smallest constant \( c \) such that for any non-negative real numbers \( x_{i} \) (where \( 1 \leq i \leq n \)), the inequality
\[ \sum_{1 \leq i < j \leq n} x_{i}x_{j}(x_{i}^{2} + x_{j}^{2}) \leq c \left( \sum_{i=1}^{n} x_{i} \right)^{4} \]
holds. Additionally, determine the necessary and sufficient conditions for equality for this constant \( c \).

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