
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 each integer \( a_{0}>1 \), define a sequence \( a_{0}, a_{1}, \cdots \) as follows: For any \( n \geqslant 0 \),
$$
a_{n+1}=\left\{\begin{array}{ll}
\sqrt{a_{n}}, & \sqrt{a_{n}} \text{ is an integer; } \\
a_{n}+3, & \text{ otherwise. }
\end{array}\right.
$$

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

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