Fun with circles

Geometry Level 3

Let \(C\) be a circle with centre \(P_0\) and \(AB\) be a diameter of \(C\). Suppose \(P_1\) is the midpoint of the line segment \(P_0B\), \(P_2\) is the midpoint of the line segment \(P_1B\) and so on. Let \(C_1, C_2, C_3 , \ldots \) be circles with diameters \(P_0P_1,P_1P_2,P_2P_3,\ldots \) respectively. Suppose the circles \( C_1, C_2, C_3, \ldots \) are all shaded. If the ratio of the area of the unshaded portion of \(C\) to that of the original \( C\) be expressed as \(M:N\), then find \(M+N\).

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