AMC 10 2021-fall-b Problem 25

A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

$(\textbf{A})\: 14 (\textbf{B}) \: 23 (\textbf{C}) \: 46 (\textbf{D}) \: 59 (\textbf{E}) \: 67$