Circle

Let \(M = \{ (x,y) \in R \times R:{x^2} + {y^2} \le {r^2}\}\), where r > 0. Consider the geometric progression \({a_n} = {1 \over {{2^{n - 1}}}}\), n = 1, 2, 3, ...... . Let S₀ = 0 and for n \(\ge\) 1, let S_n denote the sum of the first n terms of this progression. For n \(\ge\) 1, let C_n denote the circle with center (S_n\(-\)1, 0) and radius a_n, and D_n denote the circle with center (S_n\(-\)1, S_n\(-\)1) and radius a_n.

Consider M with \(r = {{({2^{199}} - 1)\sqrt 2 } \over {{2^{198}}}}\). The number of all those circles D_n that are inside M is