Two straight pipes (circular cylinders), with radii $1$ and $\frac{1}{4}$, lie parallel and in contact on a flat floor. The figure below shows a head-on view. What is the sum of the possible radii of a third parallel pipe lying on the same floor and in contact with both? $\textbf{(A)}~\frac{1}{9} \qquad$ $\textbf{(B)}~1 \qquad$ $\textbf{(C)}~\frac{10}{9} \qquad$ $\textbf{(D)}~\frac{11}{9} \qquad$ $\textbf{(E)}~\frac{19}{9}$