Modularity and Recursive Sequences

The sequence \(a_n\) is defined by

\[\large \begin{cases} a_0=3 \\ a_{n+1}-a_n=n(a_n-1) & \text{for } n \ge 0 \end{cases} \]

Find all positive integers \(m\) such that \(\gcd(m,a_n)=1\) for all \(n \geq 0\). What can the solution set for \(m\) be best described as?

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