A unit cube $ABCD-EFGH$ with edge length 1. Point $P$ is the midpoint of edge $\overline{CG}$. Points $Q$ and $R$ are on edges $\overline{BF}$ and $\overline{DH}$ respectively, and $A$, $Q$, $P$, $R$ are the four vertices of a parallelogram, as shown in the figure below.
A coordinate system is established such that the coordinates of $D$, $A$, $C$, $H$ are $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$ respectively, and $\overline{BQ} = t$. Answer the following questions.
(1) Find the coordinates of point $P$. (2 points)
(2) Find the vector $\overrightarrow{AR}$ (express in terms of $t$). (2 points)
(3) Prove that the volume of the pyramid $G-AQPR$ is a constant (independent of $t$), and find this constant value. (4 points)
(4) When $t = \frac{1}{4}$, find the distance from point $G$ to the plane containing parallelogram $AQPR$. (4 points)

In coordinate space, $O$ is the origin, and point $P$ is in the first octant with $\overline{OP} = 1$. The line $OP$ makes an angle of $45^\circ$ with the $x$-axis, and the distance from point $P$ to the $y$-axis is $\frac{\sqrt{6}}{3}$. Select the $z$-coordinate of point $P$.
(1) $\frac{1}{2}$
(2) $\frac{\sqrt{2}}{4}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{6}}{6}$
(5) $\frac{\sqrt{3}}{6}$

In coordinate space, there is a line $L$ with direction vector $( 1 , - 2, 2 )$, plane $E _ { 1 } : 2 x + 3 y + 6 z = 10$, and plane $E _ { 2 } : 2 x + 3 y + 6 z = - 4$. The length of the line segment of $L$ cut off by $E _ { 1 }$ and $E _ { 2 }$ is . (Express as a fraction in lowest terms)