
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 which positive real numbers \(a, b\) does the inequality

\[ x_{1} \cdot x_{2} + x_{2} \cdot x_{3} + \cdots + x_{n-1} \cdot x_{n} + x_{n} \cdot x_{1} \geq x_{1}^{a} \cdot x_{2}^{b} \cdot x_{3}^{a} + x_{2}^{a} \cdot x_{3}^{b} \cdot x_{4}^{a} + \cdots + x_{n}^{a} \cdot x_{1}^{b} \cdot x_{2}^{a} \]

hold for all integers \(n > 2\) and positive real numbers \(x_{1}, x_{2}, \ldots, x_{n}\)?

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