HMMT 2018 A9

Assume the quartic $x^{4}-a x^{3}+b x^{2}-a x+d=0$ has four real roots $\frac{1}{2} \leq x_{1}, x_{2}, x_{3}, x_{4} \leq 2$. Find the maximum possible value of $\frac{\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right) x_{4}}{\left(x_{4}+x_{2}\right)\left(x_{4}+x_{3}\right) x_{1}}$ (over all valid choices of $\left.a, b, d\right)$.