
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 the inequality

\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]

holds for any  $x_{1}, \ldots ,x_{n} \geq 0$  (the sum on the left consists of  $\binom{n}{2}$  summands). For this constant  $C$ , characterize the instances of equality.

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