
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 positive integer \( n \geq 3 \), for an \( n \)-element real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\), if every permutation \( y_{1}, y_{2}, \cdots, y_{n} \) of it satisfies \(\sum_{i=1}^{n-1} y_{i} y_{i+1} \geq -1\), then the real array \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) is called "glowing". Find the largest constant \( C = C(n) \) such that for every glowing \( n \)-element real array, \(\sum_{1 \leq i < j \leq n} x_{i} x_{j} \geq C \).

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