
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, \( n \geqslant 2 \).

(a) Determine the minimal constant \( c \) such that the inequality
$$
\sum_{1 \leqslant i < j \leqslant n} x_i x_j \left(x_i^2 + x_j^2\right) \leqslant c \left( \sum_{1 \leqslant i \leqslant n} x_i \right)^4
$$
holds for all non-negative real numbers \( x_1, x_2, \cdots, x_n \geqslant 0 \).

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

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