Coordinate Geometry: JEE Advanced 2021 Mathematics question

Let \[\mathrm{M}=\left\{(\mathrm{x}, \mathrm{y}) \in \mathbb{R} \times \mathbb{R}: \mathrm{x}²+\mathrm{y}² \leq \mathrm{r}²\right\}\] where $\mathrm{r}>0$. Consider the geometric progression $a_n=\frac{1}{2^{n-1}}, n=1,2,3, \ldots$. Let $S_0=0$ and, for $n \geq 1$, let $S_n$ denote the sum of the first $n$ terms of this progression. For $n \geq 1$, let $C_n$ denote the circle with center $\left(S_{n-1}, 0\right)$ and radius $a_n$, and $D_n$ denote the circle with center $\left(S_{n-1}, S_{n-1}\right)$ and radius $a_n$. Consider $M$ with $r=\frac{1025}{513}$. Let $k$ be the number of all those circles $C_{n}$ that are inside $M$. Let $l$ be the maximum possible number of circles among these $k$ circles such that no two circles intersect. Then