As shown in the figure, there is a right triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ with $\overline { \mathrm { OA } _ { 1 } } = 4$ and $\overline { \mathrm { OB } _ { 1 } } = 4 \sqrt { 3 }$. Let $\mathrm { B } _ { 2 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 1 } }$ meets the line segment $\mathrm { OB } _ { 1 }$. The figure $R _ { 1 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ but outside the sector $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 2 }$.\\
In figure $R _ { 1 }$, let $\mathrm { A } _ { 2 }$ be the point where the line passing through $\mathrm { B } _ { 2 }$ and parallel to segment $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 }$ meets segment $\mathrm { OA } _ { 1 }$, and let $\mathrm { B } _ { 3 }$ be the point where the circle with center O and radius $\overline { \mathrm { OA } _ { 2 } }$ meets segment $\mathrm { OB } _ { 2 }$. The figure $R _ { 2 }$ is obtained by shading the region that is inside triangle $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 2 }$ but outside the sector $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]\\
\includegraphics[max width=\textwidth, alt={}, center]{26f53191-c2e9-4115-98d0-f3837666a14d-06_479_290_1087_1157}\\
\includegraphics[max width=\textwidth, alt={}, center]{26f53191-c2e9-4115-98d0-f3837666a14d-06_479_359_1089_1481}\\
(1) $\frac { 3 } { 2 } \pi$\\
(2) $\frac { 5 } { 3 } \pi$\\
(3) $\frac { 11 } { 6 } \pi$\\
(4) $2 \pi$\\
(5) $\frac { 13 } { 6 } \pi$