As shown in the figure, there is a circle $O$ with diameter AB of length 4. Let C be the center of the circle, and let D and P be the midpoints of segments AC and BC, respectively. Let E and Q be the points where the perpendicular bisector of segment AC and the perpendicular bisector of segment BC meet the upper semicircle of circle $O$, respectively. Draw a square DEFG with side DE that meets circle $O$ at point A and has diagonal DF, and draw a square PQRS with side PQ that meets circle $O$ at point B and has diagonal PR. Color the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square DEFG, and the $\square$-shaped figure that is the common part of the interior of circle $O$ and the interior of square PQRS to obtain figure $R _ { 1 }$. In figure $R _ { 1 }$, draw circle $O _ { 1 }$ centered at point F with radius $\frac { 1 } { 2 } \overline { \mathrm { DE } }$, and circle $O _ { 2 }$ centered at point R with radius $\frac { 1 } { 2 } \overline { \mathrm { PQ } }$. Color 2 $\square$-shaped figures and 2 $\square$-shaped figures created in the same way as obtaining figure $R _ { 1 }$ on the two circles $O _ { 1 }$ and $O _ { 2 }$ to obtain figure $R _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in figure $R _ { n }$ obtained the $n$-th time. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 12 \pi - 9 \sqrt { 3 } } { 10 }$
(2) $\frac { 8 \pi - 6 \sqrt { 3 } } { 5 }$
(3) $\frac { 32 \pi - 24 \sqrt { 3 } } { 15 }$
(4) $\frac { 28 \pi - 21 \sqrt { 3 } } { 10 }$
(5) $\frac { 16 \pi - 12 \sqrt { 3 } } { 5 }$