As shown in the figure, there is a circle O with diameter AB of length 2. Let C be one of the two points where the line passing through the center of circle O and perpendicular to segment AB meets the circle. The figure obtained by shading the region that is outside the circle centered at C passing through points A and B and inside circle O is called $R_1$. In figure $R_1$, circles are inscribed in each of the 2 quarter-circles obtained by bisecting the semicircle of circle O that does not include the shaded part. In these 2 circles, 2 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_2$. In figure $R_2$, circles are inscribed in each of the 4 quarter-circles obtained by bisecting the semicircles of the 2 newly created circles that do not include the shaded parts. In these 4 circles, 4 triangular figures are created using the same method as for figure $R_1$ and shaded to obtain figure $R_3$. Continuing this process, let $S_n$ be the area of the shaded part in the $n$-th figure $R_n$. What is the value of $\lim_{n \rightarrow \infty} S_n$? [4 points]
(1) $\frac{5 + 2\sqrt{2}}{7}$
(2) $\frac{5 + 3\sqrt{2}}{7}$
(3) $\frac{5 + 4\sqrt{2}}{7}$
(4) $\frac{5 + 5\sqrt{2}}{7}$
(5) $\frac{5 + 6\sqrt{2}}{7}$