
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, \( n \geq 2 \).
(I) Find the minimum constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j \left( x_i^2 + x_j^2 \right) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds.
(II) For this constant \( c \), determine the necessary and sufficient conditions for which equality holds.

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