
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
A bounded sequence \( x_{0}, x_{1}, x_{2}, \ldots \) such that for all natural numbers \( i \) and \( j \), where \( i \neq j \), the following inequality holds:
\[ \left|x_{i} - x_{j}\right| |i - j|^{a} \geq 1 \]

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