In coordinate space, on the plane $x - y + 2 z = 3$ there are two distinct lines $L : \frac { x } { 2 } - 1 = y + 1 = - 2 z$ and $L ^ { \prime }$ . It is known that $L$ also lies on another plane $E$ , and the projection of $L ^ { \prime }$ on $E$ coincides with $L$ . Then the equation of $E$ is $x +$ (16-1)(16-2) $y +$ (16-3)(16-4) $z =$ (16-5) .

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 $x + ay + bz = c$).