HMMT November 2021 E10

Real numbers $x, y, z$ satisfy

$$x+x y+x y z=1, y+y z+x y z=2, z+x z+x y z=4$$

The largest possible value of $x y z$ is $\frac{a+b \sqrt{c}}{d}$, where $a, b, c, d$ are integers, $d$ is positive, $c$ is square-free, and $\operatorname{gcd}(a, b, d)=1$. Find $1000 a+100 b+10 c+d$.