Will you integrate? (2)

Calculus Level 5

Let \({V}_{1}\) be the volume generated by all the possible positions of \(P(x,y,z)\) such that \[\left\lfloor \left| x \right| \right\rfloor +\left\lfloor \left| y \right| \right\rfloor +\left\lfloor \left| z \right| \right\rfloor =n\] And let \({V}_{2}\) be the volume enclosed by the surface \[\left| x \right| +\left| y \right| +\left| z \right| =\frac{n+1}{\sqrt{2}}\] If \({V}_{1}+{V}_{2} =800 \text{ unit}^{3}\). Find \(n\)

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