There is a circle with radius 1. As shown in the figure, a rectangle with the ratio of width to height of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$