
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 \( n \geq 2 \) be a fixed integer. Find the least constant \( C \) such that the inequality
\[ \sum_{i<j} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leq C\left(\sum_{i} x_{i}\right)^{4} \]
holds for every \( x_{1}, \ldots, x_{n} \geq 0 \) (the sum on the left consists of \(\binom{n}{2}\) summands). For this constant \( \bar{C} \), characterize the instances of equality.

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