The below object's surface area and the volume defined by the region lying inside the cylinder \(x^2 + y^2 = 1\) and inside the sphere \((x - 1)^2 + y^2 + z^2 = 4\) can be represented by

\[ \pi \cdot A^3 \quad \text{ and } \quad \pi \cdot \dfrac{A^4}B - \dfrac{A^6}{B^2} , \]

respectively, where \(A\) and \(B\) are coprime positive integers.

Find \(A+B\).

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