
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
For any integer \( a_{0}>1 \), define the sequence \( a_{0}, a_{1}, \ldots \) by \( a_{n+1}=\sqrt{a_{n}} \) if \( \sqrt{a_{n}} \) is an integer, and \( a_{n+1}=a_{n}+3 \) otherwise for all \( n \geqslant 0 \).

Determine all values of \( a_{0} \) for which there exists a number \( A \) such that \( a_{n}=A \) for infinitely many values of \( n \).

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