
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 positive real numbers \( a \) and \( b \) with \( a < b \), let \( x_1, x_2, \cdots, x_{2022} \in [a, b] \). Find the maximum value of \( \frac{\left|x_1 - x_2\right| + \left|x_2 - x_3\right| + \cdots + \left|x_{2021} - x_{2022}\right| + \left|x_{2022} - x_1\right|}{x_1 + x_2 + \cdots + x_{2022}} \).

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