Let \({A}_{1}\) be the area of polygon with vertices \(P (x,y)\) such that \(\left\lfloor \left| x \right| \right\rfloor +\left\lfloor \left| y \right| \right\rfloor =n\); and \({A}_{2}\) be the area bounded by the curve \(\left| x \right| +\left| y \right| =n+1\). Find value of \(n\) such that \(\left|{A}_{1}-{A}_{2}\right|=96 \text{ unit}^2 \).

**Bonus**: Generalize for all positive integers \(n\).

**Notations**:

\( \lfloor \cdot \rfloor \) denotes the floor function.

\( | \cdot | \) denotes the absolute value function.

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