[Calculus] The position $( x , y )$ of a point P moving on the coordinate plane at time $t$ is given by $$\left\{ \begin{array} { l } x = 4 ( \cos t + \sin t ) \\ y = \cos 2 t \end{array} \quad ( 0 \leqq t \leqq 2 \pi ) \right.$$ When the distance traveled by point P from $t = 0$ to $t = 2 \pi$ is $a \pi$, find the value of $a ^ { 2 }$. [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]

The coordinates $( x , y )$ of a moving point P are given by the following functions in time $t$:
$$\begin{aligned} & x = 4 t - \sin 4 t \\ & y = 4 - \cos 4 t \end{aligned}$$
(1) The derivatives of $x$ and $y$ with respect to $t$ are
$$\begin{aligned} \frac { d x } { d t } & = \mathbf { A } ( \mathbf { A } - \mathbf { B } \cos 4 t ) \\ \frac { d y } { d t } & = \mathbf { C } \sin 4 t . \end{aligned}$$
Hence we have
$$\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \mathbf { D E } \sin ^ { 2 } \mathbf { F } t$$
(2) As the point P moves from the time $t = 0$ to the time $t = 2 \pi$, its speed $v$ is maximized a total of $\mathbf { G }$ times. Let us denote by $t _ { 0 }$ the moment of the first time the speed is maximized and the moment of the last time it is maximized by $t _ { 1 }$. Then
$$t _ { 0 } = \frac { \mathbf { H } } { \mathbf { I } } \pi , \quad t _ { 1 } = \frac { \mathbf { J } } { \mathbf { I } } \pi$$
and the maximum speed is $v = \mathbf { L }$.
(3) For $t _ { 0 }$ and $t _ { 1 }$ in (2), the distance that point P moves during the period from $t = t _ { 0 }$ to $t = t _ { 1 }$ is $\mathbf{MN}$.

In the three-dimensional orthogonal coordinate system $x y z$, the unit vectors along the $x , y$, and $z$ directions are $\mathbf { i } , \mathbf { j }$, and $\mathbf { k }$, respectively. Using the parameter $\theta ( 0 \leq \theta \leq \pi )$, we define two curves by their vector functions $\mathbf { P } ( \theta )$ and $\mathbf { Q } ( \theta )$ :
$$\begin{aligned} & \mathbf { P } ( \theta ) = x ( \theta ) \mathbf { i } + y ( \theta ) \mathbf { j } \\ & \mathbf { Q } ( \theta ) = \mathbf { P } ( \theta ) + z ( \theta ) \mathbf { k } \end{aligned}$$
where
$$\begin{aligned} & x ( \theta ) = \frac { 3 } { 2 } \cos ( \theta ) - \frac { 1 } { 2 } \cos ( 3 \theta ) \\ & y ( \theta ) = \frac { 3 } { 2 } \sin ( \theta ) - \frac { 1 } { 2 } \sin ( 3 \theta ) \end{aligned}$$
Here, $z ( \theta )$ is a continuous function satisfying $z ( 0 ) > 0$ and $z ( \pi ) < 0$, and the curve parametrized by $\mathbf { Q } ( \theta )$ lies on the sphere of radius 2, centered at the origin $( 0,0,0 )$ of the coordinate system. The positive direction of a curve corresponds to increasing values of the parameter $\theta$. Note that the curvature is the reciprocal of the radius of curvature. Answer the following questions.
I. As $\theta$ is varied from 0 to $\pi$, calculate the arc length of the curve parametrized by $\mathbf { P } ( \theta )$.
II. Obtain $z ( \theta )$.
III. Let $\alpha$ be the angle between the tangent of the curve parametrized by $\mathbf { Q } ( \theta )$ and the unit vector $\mathbf { k }$. Calculate $\cos ( \alpha )$.
IV. Find the curvature $\kappa _ { P } ( \theta )$ of the curve parametrized by $\mathbf { P } ( \theta )$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.
V. Let $\kappa _ { Q } ( \theta )$ be the curvature of the curve parametrized by $\mathbf { Q } ( \theta )$. Express $\kappa _ { Q } ( \theta )$ in terms of $\kappa _ { P } ( \theta )$ and $\alpha$. Here, $\theta = 0$ and $\theta = \pi$ are excluded.