As shown in the figure, there is a semicircle with diameter AB of length 2. Two points $\mathrm { P } , \mathrm { Q }$ are taken on arc AB such that $\angle \mathrm { PAB } = \theta , \angle \mathrm { QBA } = 2 \theta$, and the intersection of two line segments $\mathrm { AP } , \mathrm { BQ }$ is denoted R. Points S on segment AB, point T on segment BR, and point U on segment AR are chosen such that segment UT is parallel to segment AB and triangle STU is equilateral. Let $f ( \theta )$ be the area of the region enclosed by two line segments $\mathrm { AR } , \mathrm { QR }$ and arc AQ, and let $g ( \theta )$ be the area of triangle STU. When $\lim _ { \theta \rightarrow 0 + } \frac { g ( \theta ) } { \theta \times f ( \theta ) } = \frac { q } { p } \sqrt { 3 }$, find the value of $p + q$. (Given that $0 < \theta < \frac { \pi } { 6 }$ and $p$ and $q$ are coprime natural numbers.) [4 points]
As shown in the figure, there is a semicircle with diameter AB of length 2. Two points $\mathrm { P } , \mathrm { Q }$ are taken on arc AB such that $\angle \mathrm { PAB } = \theta , \angle \mathrm { QBA } = 2 \theta$, and the intersection of two line segments $\mathrm { AP } , \mathrm { BQ }$ is denoted R. Points S on segment AB, point T on segment BR, and point U on segment AR are chosen such that segment UT is parallel to segment AB and triangle STU is equilateral. Let $f ( \theta )$ be the area of the region enclosed by two line segments $\mathrm { AR } , \mathrm { QR }$ and arc AQ, and let $g ( \theta )$ be the area of triangle STU. When $\lim _ { \theta \rightarrow 0 + } \frac { g ( \theta ) } { \theta \times f ( \theta ) } = \frac { q } { p } \sqrt { 3 }$, find the value of $p + q$.
(Given that $0 < \theta < \frac { \pi } { 6 }$ and $p$ and $q$ are coprime natural numbers.) [4 points]