Question 57
Real numbers $x, y, z$ satisfy
$$
x + x y + x y z = 1, \quad y + y z + x y z = 2, \quad z + x z + x y z = 4
$$
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 $1000 a + 100 b + 10 c + d$.
$$
x + x y + x y z = 1, \quad y + y z + x y z = 2, \quad z + x z + x y z = 4
$$
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 $1000 a + 100 b + 10 c + d$.