View source for 2021 Fall AMC 10B Problems/Problem 25

A rectangle with side lengths <math>1{ }</math> and <math>3,</math> a square with side length <math>1,</math> and a rectangle <math>R</math> are inscribed inside a larger square as shown. The sum of all possible values for the area of <math>R</math> can be written in the form <math>\tfrac mn</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m+n?</math>
[asy]
size(8cm);
draw((0,0)--(10,0));
draw((0,0)--(0,10));
draw((10,0)--(10,10));
draw((0,10)--(10,10));
draw((1,6)--(0,9));
draw((0,9)--(3,10));
draw((3,10)--(4,7));
draw((4,7)--(1,6));
draw((0,3)--(1,6));
draw((1,6)--(10,3));
draw((10,3)--(9,0));
draw((9,0)--(0,3));
draw((6,13/3)--(10,22/3));
draw((10,22/3)--(8,10));
draw((8,10)--(4,7));
draw((4,7)--(6,13/3));
label("<math>3</math>",(9/2,3/2),N);
label("<math>3</math>",(11/2,9/2),S);
label("<math>1</math>",(1/2,9/2),E);
label("<math>1</math>",(19/2,3/2),W);
label("<math>1</math>",(1/2,15/2),E);
label("<math>1</math>",(3/2,19/2),S);
label("<math>1</math>",(5/2,13/2),N);
label("<math>1</math>",(7/2,17/2),W);
label("<math>R</math>",(7,43/6),W);
[/asy]
<math>(\textbf{A})\: 14\qquad(\textbf{B}) \: 23\qquad(\textbf{C}) \: 46\qquad(\textbf{D}) \: 59\qquad(\textbf{E}) \: 67</math>