\((\alpha_1, \beta_1), \ (\alpha_2, \beta_2), \ \cdots \ , \ (\alpha_n, \beta_n)\) are the points on the curve \[S \ : \ y=\sin (x+y) \ \ \forall x \in [-4\pi, \ 4\pi]\]

such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]

such that the tangents at these points to \(S\) are perpendicular to the line \[l \ : \ (\sqrt {2}-1)x+y=0.\]

Find the value of \[\left\lfloor \displaystyle\sum_{k=1}^{n} |\alpha_k|-|\beta_k|\right\rfloor.\]

**Details and Assumptions**

\(\lfloor{\cdots}\rfloor\) denotes the floor function (greatest integer function).

Even though in the problem, a plural form of points is given, there may only be one such point that exists.

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