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 $a+b+c$? $\textbf{(A)} ~10\qquad$ $\textbf{(B)} ~11\qquad$ $\textbf{(C)} ~12\qquad$ $\textbf{(D)} ~13\qquad$ $\textbf{(E)} ~14$