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)
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)