I am trying to solve the following over determined system of polynomials

\begin{align} & p_1(x_1,x_2,\ldots,x_n)=0, \\ & p_2(x_1,x_2,\ldots,x_n)=0, \\ & \vdots \\ & p_m(x_1,x_2, \ldots, x_n)=0, \\ & (x_1-l_1) (x_1-(l_1+1)) (x_1 - (l_1 + 2))\cdots (x_1-(u_1-1))(x_1-u_1)=0, \\ & \vdots \\ & (x_n-l_n) (x_n-(l_n+1)) (x_n - (l_n + 2))\cdots (x_n-(u_n-1))(x_n-u_n)=0 \cdots (x_n-u_n)=0. \end{align}

Here $l_1,u_1, \ldots, l_n, u_n$ are non-negative integers. The last $n$ equations mean that $x_i \in \{l_i, l_i +1, l_i+2, \ldots, u_i-1, u_i\}$ for all $i \in \{1,\ldots, n\}$. I know that we can use Groebner bases to solve the polynomial system.

But is there a better way to solve such over-determined polynomial system? I will appreciate any suggestion or direction.