
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 a fixed integer \( n \) where \( n \geq 2 \):

a) Determine the smallest constant \( c \) such that 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 for all nonnegative real numbers \( x_1, x_2, \ldots, x_n \geq 0 \).

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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