
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]


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Given a real number \( a > 1 \), construct a bounded infinite sequence \( x_{0}, x_{1}, x_{2}, \cdots \) such that for every pair of distinct non-negative integers \( i, j \), the inequality \( \left|x_{i}-x_{j}\right| \cdot |i-j|^{a} \geqslant 1 \) holds.

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