
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
Let \( n \) be a fixed integer with \( n \geq 2 \).

(1) Determine the smallest constant \( c \) such that the inequality
\[ 
\sum_{1 \leq i < j \leq n} x_{i} x_{j} \left(x_{i}^{2} + x_{j}^{2} \right) \leqslant c \left( \sum_{1 \leq i \leq n} x_{i} \right)^{4}
\]
holds for all non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \).

(2) For this constant \( c \), determine the necessary and sufficient condition for the equality to hold.

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