
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
Find at least one real number  $A{}$  such that for any positive integer  $n{}$  the distance between  $\lceil A^n\rceil$  and the nearest square of an integer is equal to two. 

*Dmitry Krekov*

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