In coordinate space, consider three planes $E_{1}: x + y + z = 7$, $E_{2}: x - y + z = 3$, $E_{3}: x - y - z = -5$. Let $L_{3}$ be the line of intersection of $E_{1}$ and $E_{2}$; $L_{1}$ be the line of intersection of $E_{2}$ and $E_{3}$; $L_{2}$ be the line of intersection of $E_{3}$ and $E_{1}$. If a fourth plane $E_{4}$ together with $E_{1}, E_{2}, E_{3}$ encloses a regular tetrahedron with edge length $6\sqrt{2}$, find the equation of $E_{4}$ (write in the form $x + ay + bz = c$).

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
If the projection of the origin $O$ onto plane $E$ is point $Q$, and the angle between vector $\overrightarrow{OQ}$ and vector $(1, 0, 0)$ is $\alpha$, what is the value of $\cos\alpha$? (Single choice question, 3 points)
(1) $-\frac{\sqrt{2}}{2}$
(2) $-\frac{1}{2}$
(3) $\frac{1}{2}$
(4) $\frac{\sqrt{2}}{2}$
(5) $\frac{\sqrt{3}}{2}$

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
It is known that there is a point $P(a, b, c)$ in space such that the angle $\theta$ between vector $\overrightarrow{OP}$ and vector $(1, 0, 0)$ satisfies $\theta \leq \frac{\pi}{6}$. Show that the real numbers $a, b, c$ satisfy the inequality $a^{2} \geq 3\left(b^{2} + c^{2}\right)$. (Non-multiple choice question, 4 points)

In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
Continuing from question 19, it is known that point $P$ is on plane $E$ and $b = 0$. Find the maximum possible range of $c$ and the minimum possible length of line segment $\overline{OP}$. (Non-multiple choice question, 8 points)

In coordinate space, point $A$ has coordinates $(a, b, c)$, where $a, b, c$ are all negative real numbers. Point $A$ is at distance 6 from each of the three planes $E _ { 1 } : 4 y + 3 z = 2$, $E _ { 2 } : 3 y + 4 z = - 5$, and $E _ { 3 } : x + 2 y + 2 z = - 2$. Then $a + b + c =$ (14-1) (14-2) (14-3).