HMMT November 2020 H9

While waiting for their next class on Killian Court, Alesha and Belinda both write the same sequence $S$ on a piece of paper, where $S$ is a 2020-term strictly increasing geometric sequence with an integer common ratio $r$. Every second, Alesha erases the two smallest terms on her paper and replaces them with their geometric mean, while Belinda erases the two largest terms in her paper and replaces them with their geometric mean. They continue this process until Alesha is left with a single value $A$ and Belinda is left with a single value $B$. Let $r_{0}$ be the minimal value of $r$ such that $\frac{A}{B}$ is an integer. If $d$ is the number of positive factors of $r_{0}$, what is the closest integer to $\log _{2} d$ ?