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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.

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