AMC 10 2021-fall-a Problem 23

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7 \textbf{(B) }8 \textbf{(C) }9 \textbf{(D) }10 \textbf{(E) }11$