Let $S$ be the set of points in the Cartesian plane that satisfy <center>$\Big|\big| |x|-2\big|-1\Big|+\Big|\big| |y|-2\big|-1\Big|=1.$</center> 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 $a+b$.