As shown in the figure, there is an equilateral triangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 }$ with side length 1. Let $\mathrm { D } _ { 1 }$ be the midpoint of segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$, and let $\mathrm { B } _ { 2 }$ be a point on segment $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } } = \overline { \mathrm { C } _ { 1 } \mathrm {~B} _ { 2 } }$. Draw a sector $\mathrm { C } _ { 1 } \mathrm { D } _ { 1 } \mathrm {~B} _ { 2 }$ with center $\mathrm { C } _ { 1 }$. Let $\mathrm { A } _ { 2 }$ be the foot of the perpendicular from $\mathrm { B } _ { 2 }$ to segment $\mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, and let $\mathrm { C } _ { 2 }$ be the midpoint of segment $\mathrm { C } _ { 1 } \mathrm {~B} _ { 2 }$. The figure $R _ { 1 }$ is obtained by shading the region enclosed by two segments $\mathrm { B } _ { 1 } \mathrm {~B} _ { 2 } , \mathrm {~B} _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { D } _ { 1 } \mathrm {~B} _ { 2 }$, and the interior of triangle $\mathrm { C } _ { 1 } \mathrm {~A} _ { 2 } \mathrm { C } _ { 2 }$. In figure $R _ { 1 }$, let $\mathrm { D } _ { 2 }$ be the midpoint of segment $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 }$, and let $\mathrm { B } _ { 3 }$ be a point on segment $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 }$ such that $\overline { \mathrm { C } _ { 2 } \mathrm { D } _ { 2 } } = \overline { \mathrm { C } _ { 2 } \mathrm {~B} _ { 3 } }$. Draw a sector $\mathrm { C } _ { 2 } \mathrm { D } _ { 2 } \mathrm {~B} _ { 3 }$ with center $\mathrm { C } _ { 2 }$. Let $\mathrm { A } _ { 3 }$ be the foot of the perpendicular from $\mathrm { B } _ { 3 }$ to segment $\mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$, and let $\mathrm { C } _ { 3 }$ be the midpoint of segment $\mathrm { C } _ { 2 } \mathrm {~B} _ { 3 }$. The figure $R _ { 2 }$ is obtained by shading the region enclosed by two segments $\mathrm { B } _ { 2 } \mathrm {~B} _ { 3 }$, $\mathrm { B } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { D } _ { 2 } \mathrm {~B} _ { 3 }$, and the interior of triangle $\mathrm { C } _ { 2 } \mathrm {~A} _ { 3 } \mathrm { C } _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the shaded part in the $n$-th figure $R _ { n }$. Find the value of $\lim _ { n \rightarrow \infty } S _ { n }$. [4 points]
(1) $\frac { 11 \sqrt { 3 } - 4 \pi } { 56 }$
(2) $\frac { 11 \sqrt { 3 } - 4 \pi } { 52 }$
(3) $\frac { 15 \sqrt { 3 } - 6 \pi } { 56 }$
(4) $\frac { 15 \sqrt { 3 } - 6 \pi } { 52 }$
(5) $\frac { 15 \sqrt { 3 } - 4 \pi } { 52 }$