Question: How to solve this system

I have a polynomial system that I'd like to find ONE solution of (there may be many, so I want the program to terminate after 1 is found, since I only need to know that there IS a solution). In this system, the unknown variables are a_i, b_i, c_i, d_i, s_i, t_i, r_i, l_i, for i = 1, 2, 3, 4. Thus I want the solution in terms of these variables. The system is given below.

\begin{align*}
q & = \sum_{i = 1}^4 a_i s_i, \\
w & = \sum_{i = 1}^4 a_i t_i, \\
y & = \sum_{i = 1}^4 a_i r_i, \\
u & = \sum_{i = 1}^4 a_i l_i, \\
&\\
p & = \sum_{i = 1}^4 b_i s_i, \\
f & = \sum_{i = 1}^4 b_i t_i, \\
g & = \sum_{i = 1}^4 b_i r_i, \\
h & = \sum_{i = 1}^4 b_i l_i, \\
&\\
j & = \sum_{i = 1}^4 c_i s_i, \\
k & = \sum_{i = 1}^4 c_i t_i, \\
z & = \sum_{i = 1}^4 c_i r_i, \\
x & = \sum_{i = 1}^4 c_i l_i, \\
&\\
v & = \sum_{i = 1}^4 d_i s_i, \\
n & = \sum_{i = 1}^4 d_i t_i, \\
m & = \sum_{i = 1}^4 d_i r_i, \\
o & = \sum_{i = 1}^4 d_i l_i,
\end{align*}
subject to the constraints $a_i d_i = b_i c_i$ and $s_i l_i = t_i r_i$ for $i = 1, 2, 3, 4$.