Inside a rectangle with width 6 and height 8, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the rectangle is drawn to obtain figure $R _ { 1 }$. From figure $R _ { 1 }$, four rectangles are drawn with each segment from a vertex of the rectangle to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 2 }$.
In figure $R _ { 2 }$, for each of the four congruent rectangles, four rectangles are drawn with each segment from a vertex to the intersection of the diagonal and circle as the diagonal. Inside each of the newly drawn rectangles, a circle with center at the intersection of the two diagonals and diameter equal to $\frac { 1 } { 3 }$ of the width of the newly drawn rectangle is drawn to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the sum of the areas of all circles in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? (Here, the widths and heights of all rectangles are parallel to each other, respectively.) [4 points]
(1) $\frac { 37 } { 9 } \pi$
(2) $\frac { 34 } { 9 } \pi$
(3) $\frac { 31 } { 9 } \pi$
(4) $\frac { 28 } { 9 } \pi$
(5) $\frac { 25 } { 9 } \pi$