19.4.3 ** For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+3, i=1,2, \cdots, n$. If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime, find the largest possible value of $n$.