
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 every integer $a_{0}>1$, we define the sequence $a_{0}, a_{1}, a_{2}, \ldots$ as follows. For every $n \geq 0$, let

$$
a_{n+1}= \begin{cases}\sqrt{a_{n}}, & \text { if } \sqrt{a_{n}} \text { is an integer, } \\ a_{n}+3 & \text { otherwise. }\end{cases}
$$

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

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