Consider a unit-sphere centered at \((x,y,z) = (0,0,0)\). In the same coordinate system, there is a circle with radius \((r = \frac{1}{2}\)) which lies in the \(xy\) plane and is centered at \((0,0,0)\).

If the circle is projected in the \(+z\) direction onto the surface of the sphere, the surface area enclosed by the resulting projection can be expressed as \( \pi (a - \sqrt{b})\).

Determine \((a+b)\)

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