
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
For any permutation \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) of \(1, 2, \cdots, n\) \((n \in \mathbf{N}_{+}, n > 2012)\), define:
$$
L=\sum_{i=1}^{n}\left|x_{i}-\sqrt{3} x_{i+1}\right|\left(x_{n+1}=x_{1}\right).
$$

Find \(L_{\text{max}}\), \(L_{\text{min}}\) and the number of permutations \(\left(x_{1}, x_{2}, \cdots, x_{n}\right)\) that attain the maximum value.

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