Here is a random choice of coefficients:

coeffs = Flatten@
  Table[{a[i] -> RandomInteger[3], b[i] -> RandomInteger[3], 
    c[i] -> RandomInteger[3], d[i] -> RandomInteger[3], 
    e[i] -> RandomInteger[3], f[i] -> RandomInteger[3], g[i] -> 1, 
    h[i] -> RandomInteger[3], ii[i] -> RandomInteger[3]}, {i, 4}]

and the corresponding equations

eqs = Table[
   Y[i] == a[i] X1 + b[i] X2 + c[i] X3 + d[i] X1 X2 + e[i] X1 X3 + 
     f[i] X2 X3 + g[i] X1^2 + h[i] X2^2 + ii[i] X3^2, {i, 4}] /. sost

You can try to solve the equations with Solve:

Solve[eqs, {X1, X2, X3}]

On my system it gives no solution in the first few attempts. It is a system of high degree.