$$
\begin{aligned}
A & =|| x-y|+x+y-2 z|+|x-y|+x+y+2 z \\
& =4 \max \{x, y, z\},
\end{aligned}
$$

where, $\max \{x, y, z\}$ represents the maximum of $x$, $y$, and $z$.
Hint: Divide into three cases based on $x \geqslant y, x \geqslant z ; y \geqslant z, y \geqslant x ; z \geqslant x$, $z \geqslant y$ to simplify $A$, and obtain $A$ as $4 x, 4 y, 4 z$ respectively.