As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]
As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP.\\
When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]