Let \(S\) be the set of points in the Cartesian plane that satisfy
$$\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.$$
If a model of \(S\) were built from wire of negligible thickness, then the total length of wire required would be \(a\sqrt{b}\) where \(a\) and \(b\) are positive integers and \(b\) is not divisible by the square of any prime number. Find \(a+b\).
(第十五届AIME1997 第13题)