
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 $n (n \geq 2)$ be a fixed integer,
(1) Determine the smallest constant $c$ such that the inequality $\sum x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq c\left(\sum^{n} x_{i}\right)^{4}$ 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 the equality to hold.

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