
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 \( n \)-dimensional real vector \( \left( x_1, x_2, \cdots, x_n \right) \), if every permutation \( y_1, y_2, \cdots, y_n \) of its elements satisfies \( \sum_{i=1}^{n-1} y_i y_{i+1} \geq -1 \), then the vector \( \left( x_1, x_2, \cdots, x_n \right) \) is called "shining". Find the largest constant \( C = C(n) \), such that for every shining \( n \)-dimensional real vector, it holds that \( \sum_{1 \leq i < j \leq n} x_i x_j \geq C \).

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