
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
An integer \( n \geq 3 \) is given. We call an \( n \)-tuple of real numbers \( (x_{1}, x_{2}, \ldots, x_{n}) \) Shiny if for each permutation \( y_{1}, y_{2}, \ldots, y_{n} \) of these numbers we have
$$
\sum_{i=1}^{n-1} y_{i} y_{i+1} - y_{1} y_{2} + y_{2} y_{3} + y_{3} y_{4} + \cdots + y_{n-1} y_{n} \geq -1.
$$

Find the largest constant \( K = K(n) \) such that
$$
\sum_{1 \leqslant i < j \leqslant n} x_{i} x_{j} \geq K
$$
holds for every Shiny \( n \)-tuple \( (x_{1}, x_{2}, \ldots, x_{n}) \).

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