
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
An infinite sequence  $ \,x_{0},x_{1},x_{2},\ldots \,$  of real numbers is said to be **bounded** if there is a constant  $ \,C\,$  such that  $ \, \vert x_{i} \vert \leq C\,$  for every  $ \,i\geq 0$ . Given any real number  $ \,a > 1,\,$  construct a bounded infinite sequence  $ x_{0},x_{1},x_{2},\ldots \,$  such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers  $ i, j$ .

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