
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, $n \geqslant 2$.
1. Determine the smallest constant $c$ such that the inequality
$$
\sum_{1 \leqslant i<j \leqslant n} x_i x_j (x_i^2 + x_j^2) \leqslant c \left(\sum_{i=1}^n x_i \right)^4
$$
holds for all non-negative real numbers $x_1, x_2, \cdots, x_n$.
2. For this constant $c$, determine the necessary and sufficient conditions for which equality holds.

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