As shown in the figure, let $\mathrm { P } _ { 1 } , \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 }$ be the five equal division points of the diagonal BD of a square ABCD with side length 5, in order from point B. Draw squares with line segments $\mathrm { BP } _ { 1 } , \mathrm { P } _ { 2 } \mathrm { P } _ { 3 } , \mathrm { P } _ { 4 } \mathrm { D }$ as diagonals respectively, and circles with line segments $\mathrm { P } _ { 1 } \mathrm { P } _ { 2 } , \mathrm { P } _ { 3 } \mathrm { P } _ { 4 }$ as diameters respectively. Color the figure-eight-shaped regions to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, let $\mathrm { Q } _ { 1 }$ be the vertex of the square with diagonal $\mathrm { P } _ { 2 } \mathrm { P } _ { 3 }$ that is closest to point A, and let $\mathrm { Q } _ { 2 }$ be the vertex closest to point C. Draw a square with diagonal $\mathrm { AQ } _ { 1 }$ and a square with diagonal $\mathrm { CQ } _ { 2 }$, and in these 2 newly drawn squares, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 1 }$, and color them to obtain figure $R _ { 2 }$. In figure $R _ { 2 }$, in the square with diagonal $\mathrm { AQ } _ { 1 }$ and the square with diagonal $\mathrm { CQ } _ { 2 }$, draw figure-eight-shaped regions using the same method as for obtaining figure $R _ { 2 }$ from figure $R _ { 1 }$, and color them to obtain figure $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored region in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 24 } { 17 } ( \pi + 3 )$
(2) $\frac { 25 } { 17 } ( \pi + 3 )$
(3) $\frac { 26 } { 17 } ( \pi + 3 )$
(4) $\frac { 24 } { 17 } ( 2 \pi + 1 )$
(5) $\frac { 25 } { 17 } ( 2 \pi + 1 )$