
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 the sequence  $a_0, a_1, a_2, \ldots$  for  $n \geq 0$  as  $$ a_{n+1} = 
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} 
a_n + 3 & \text{otherwise.}
\end{cases} $$  Determine all values of  $a_0$  such that there exists a number  $A$  such that  $a_n = A$  for infinitely many values of  $n$ .

*Proposed by Stephan Wagner, South Africa*

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