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 \(a + b\)?

###### This is a problem from the 2015 AMC (12 A #21).