$3 \cdot 68$ For any positive integer $q_{0}$, consider the sequence $q_{1}, q_{2}, \cdots, q_{n}$ defined by $q_{i}=\left(q_{i-1}-1\right)^{3}+3 \quad(i=1$, $2, \cdots, n)$. If each $q_{i}(i=1,2, \cdots, n)$ is a power of a prime. Find the largest possible value of $n$.
（Hungarian Mathematical Olympiad, 1990)