AMC 12 2015a Problem 21

A circle of radius $r$ passes through both foci of, and exactly four points on, the ellipse with equation $x^2+16y^2=16$. The set of all possible values of $r$ is an interval $[a,b)$. What is $a+b$?

\textbf{(A)}\ 5\sqrt{2}+4 \textbf{(B)}\ \sqrt{17}+7 \textbf{(C)}\ 6\sqrt{2}+3 \textbf{(D)}\ \sqrt{15}+8 \textbf{(E)}\ 12