In the language of Calculus, the area between two curves is essentially a difference of integrals. The applications of this calculation range from macroeconomic tax models to signal analysis.

A square has vertices at \((1,1), (-1,1), (-1,-1), \text{ and } (1,-1).\)

Let \(S\) be the region of all points inside the square which are nearer to the origin than to any edge. What is the area of \(S\)?

\(\text{Details and Assumptions:}\)

\([\mathbf{N}]\) means the area of the locus \(\mathbf{N}\), and \([PQRS]\) means the area of \(PQRS\). \(\lfloor x \rfloor\) is the floor function.

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