As shown in the figure, there is a circle O with diameter AB of length 2. A line passing through the center of circle O and perpendicular to line segment AB intersects the circle at two points, one of which is C.
A circle centered at C passing through points A and B is drawn. The region that is outside this circle and inside circle O is colored to form a triangular shape, creating figure $R _ { 1 }$. The semicircle of circle O that does not include the colored part is divided into 2 quarter circles, and circles inscribed in each quarter circle are drawn. Inside these 2 circles, two triangular shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 2 }$. The semicircles of the 2 newly created circles in figure $R _ { 2 }$ that do not include the colored parts are each divided into 2 quarter circles, and circles inscribed in each of the 4 quarter circles are drawn. Inside these 4 circles, 4 shapes are created using the same method as for figure $R _ { 1 }$ and colored, creating figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained at the $n$-th step. 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) $\frac { 5 + 6 \sqrt { 2 } } { 7 }$