# 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 $$\dfrac1n \left( {\dfrac{S}{\sqrt{2}}+1}\right)$$.

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