GAIC Math 2024 Problem 14

For any positive integer $q_0$, consider a sequence $q_1,q_2,\cdots,q_n$ defined by $q_i=\left(q_{i-1}-1\right)^3+3\left(i=1,2,\cdots,n\right)$. If every $q_i\left(i=1,2,\cdots,n\right)$ is a power of prime, then the maximum possible value of $n$ is \underline{\hspace{2cm}}.