
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
The sequence \( a_{0}, a_{1}, a_{2}, \ldots \) of positive integers satisfies
\[ 
a_{n+1} = 
\begin{cases} 
\sqrt{a_{n}}, & \text{if } \sqrt{a_{n}} \text{ is an integer} \\
a_{n} + 3, & \text{otherwise} 
\end{cases} \quad \text{ for every } n \geqslant 0.
\]
Determine all values of \( a_{0} > 1 \) for which there is at least one number \( a \) such that \( a_{n} = a \) for infinitely many values of \( n \).

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