So I want to solve for a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r which satisfy the following system of equations: ( I only need positive integer (or 0) solution)

a g + c h + b i + g j + i k + h l == 2,

b g + a h + c i + h j + g k + i l == 2,

c g + b h + a i + i j + h k + g l == 2,

d g + f h + e i + j^2 + 2 k l == 4,

e g + d h + f i + 2 j k + l^2 == 3,

f g + e h + d i + k^2 + 2 j l == 3,

a^2 + 2 b c + d g + f h + e i == 4,

2 a b + c^2 + e g + d h + f i == 3,

a d + c e + b f + d j + f k + e l == 2,

b d + a e + c f + e j + d k + f l == 2,

c d + b e + a f + f j + e k + d l == 2,

g m + i n + h o + m p + o q + n r == 2,

h m + g n + i o + n p + m q + o r == 2,

i m + h n + g o + o p + n q + m r == 2,

j m + l n + k o + p^2 + 2 q r == 4,

k m + j n + l o + 2 p q + r^2 == 3,

l m + k n + j o + q^2 + 2 p r == 3,

a + b + c + d + e + f == 4,

g + h + i + j + k + l == 4,

m + n + o + p + q + r == 4

I know that there is a solution (1,1,1,1,0,0,1,0,0,1,1,1,1,1,1,1,0,0). Now I want to find all integer solutions to see if this is unique. I tried to use Mathematica to find solutions, but it takes forever to run it, and it never gave me answer. ( I guess it is because I don't know how to put constraint as conditions ). I wonder if there is anyway to prove the solution is unique without relying on software ?