As shown in the figure, a sector $\mathrm { OA } _ { 1 } \mathrm {~B} _ { 1 }$ with radius equal to the line segment $\mathrm { OA } _ { 1 } ( 0,8 )$ connecting the origin $O$ and point $\mathrm { A } _ { 1 } ( 0,8 )$ and central angle $\theta$ is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 1 }$ to the $x$-axis is $\mathrm { A } _ { 2 }$, and a sector $\mathrm { OA } _ { 2 } \mathrm {~B} _ { 2 }$ with radius equal to segment $\mathrm { OA } _ { 2 }$ and central angle $\theta$ is drawn. The foot of the perpendicular from point $\mathrm { B } _ { 2 }$ to the $y$-axis is $\mathrm { A } _ { 3 }$, and a sector $\mathrm { OA } _ { 3 } \mathrm {~B} _ { 3 }$ with radius equal to segment $\mathrm { OA } _ { 3 }$ and central angle $\theta$ is drawn. Continuing this process of alternately dropping perpendiculars to the $x$-axis and $y$-axis in the clockwise direction, let $l _ { n }$ be the length of arc $\mathrm { A } _ { n } \mathrm {~B} _ { n }$ of sector $\mathrm { OA } _ { n } \mathrm {~B} _ { n }$. When $\sum _ { n = 1 } ^ { \infty } l _ { n } = 12 \theta$, what is the value of $\sin \theta$? (Here, $0 < \theta < \frac { \pi } { 2 }$.) [4 points]
(1) $\frac { 1 } { 7 }$
(2) $\frac { 1 } { 6 }$
(3) $\frac { 1 } { 5 }$
(4) $\frac { 1 } { 4 }$
(5) $\frac { 1 } { 3 }$