Seriously, Who Developed The Mod Function?

Geometry Level 5

The length of the graph of \(|||x|-a|-b|+|||y|-a|-b|=1 \) can take \(n\) possible values for positive integers \(a\) and \(b\). Let the sum of all these values be \(S\).

Find the value of \(\dfrac1n \left( {\dfrac{S}{\sqrt{2}}+1}\right) \).


Inspiration.

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