Infinite Cones, Finite Surface Area?

Geometry Level 3

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A unit circle is divided into halves. One of these pieces is divided into two halves, and one of those pieces is divided into two halves, and so on. Each of these sectors of the unit circle is rolled up into a cone-shape, and a circular base is capped onto each of the cones. The total surface area \(S\) of all of these cones combined can be expressed as \[S=\dfrac{a}{b}\pi\] for positive coprime integers \(a,b\). What is \(a+b\)?

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