2021 Fall AMC 10A Problem 19

A disk of radius 1 rolls all the way around the inside of a square of side length s>4 and sweeps out a region of area A. A second disk of radius 1 rolls all the way around the outside of the same square and sweeps out a region of area 2A. The value of s can be written as a+\frac{b\pi}{c}, where a,b, and c are positive integers and b and c are relatively prime. What is a+b+c?

\textbf{(A)} ~10\qquad\textbf{(B)} ~11\qquad\textbf{(C)} ~12\qquad\textbf{(D)} ~13\qquad\textbf{(E)} ~14

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.