Question 13
Let $S$ be the set of points in the Cartesian plane that satisfy $$
\begin{aligned} &\Bigl|\bigl||x| - 2\bigr| - 1\Bigr| \\ &\quad {}+ \Bigl|\bigl||y| - 2\bigr| - 1\Bigr| = 1. \end{aligned}
$$ 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 $a + b.$
\begin{aligned} &\Bigl|\bigl||x| - 2\bigr| - 1\Bigr| \\ &\quad {}+ \Bigl|\bigl||y| - 2\bigr| - 1\Bigr| = 1. \end{aligned}
$$ 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 $a + b.$