
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
Given a [rational number](https://artofproblemsolving.com/wiki/index.php/Rational_number), write it as a [fraction](https://artofproblemsolving.com/wiki/index.php/Fraction) in lowest terms and calculate the product of the resulting [numerator](https://artofproblemsolving.com/wiki/index.php/Numerator) and [denominator](https://artofproblemsolving.com/wiki/index.php/Denominator). For how many rational numbers between $0$ and $1$ will $20_{}^{}!$ be the resulting [product](https://artofproblemsolving.com/wiki/index.php/Product)?

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