[(1)] Show that when a positive integer $n$ is a multiple of $3$, $a_n$ is a multiple of $5$.
[(2)] Let $k$, $n$ be positive integers. Express the necessary and sufficient condition for $a_n$ to be a multiple of $a_k$, in terms of $k$ and $n$.
[(3)] Find the greatest common divisor of $a_{2022}$ and $(a_{8091})^2$.
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\textbf{2}
Define the sequence $\{a_n\}$ as follows.
$$a_1 = 1, \quad a_{n+1} = a_n^2 + 1 \quad (n = 1,\ 2,\ 3,\ \cdots)$$
\begin{enumerate}
\item[(1)] Show that when a positive integer $n$ is a multiple of $3$, $a_n$ is a multiple of $5$.
\item[(2)] Let $k$, $n$ be positive integers. Express the necessary and sufficient condition for $a_n$ to be a multiple of $a_k$, in terms of $k$ and $n$.
\item[(3)] Find the greatest common divisor of $a_{2022}$ and $(a_{8091})^2$.
\end{enumerate}
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