
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 \( A \) with the property that for any natural number \( n \), the distance from the ceiling of the number \( A \wedge n \) to the nearest square of an integer is 2. (The ceiling of \( x \) is the smallest integer not less than \( x \)).

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