
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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 \( \log_{2} d \)?

Present the answer in LaTex format: \boxed{Your answer}
user