Solutions to system using Fourier-Motzkin Elimination

Question

I am trying to find a solution to this system using Fourier-Motzkin Elimination, but I don't know how to finish this. Here is what I have so far.
$x_1-x_2\leq 0,\quad x_1-x_3\leq 0,\quad -2x_1+2x_2+4x_3\leq 0,\quad -x_3\leq -1$
$\iff\begin{cases} x_1-x_2\quad\qquad\leq 0 \\ x_1\quad\qquad-x_3\leq0 \\ -x_1+x_2+2x_3\leq0 \\ \quad\;\qquad\quad-x_3\leq -1 \end{cases} $
$\implies x_1\leq x_2,\quad x_1\leq x_3,\quad x_2+2x_3\leq x_1$
$\iff x_2+x_3\leq x_2,\quad x_2+2x_3\leq x_3$
$\iff 2x_3 \leq 0,\quad x_2+x_3\leq 0$
The new system should be:
$\qquad\begin{cases} x_2+x_3\leq 0 \\ \;\; \quad 2x_3\leq0 \\ \;\quad-x_3\leq -1 \end{cases}$
But there's only one inequality with $x_2$, so I can't eliminate that variable and I also have $2x_3\leq 0$ and $-x_3\leq -1$ which looks wrong. Could someone please explain to me what I did wrong? Thanks