How to Invert functions with same input variables?

Questions
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1

Dear Sirs, I want to invert following set of equations:

Y(i)=a(i)X1+b(i)X2+c(i)X3+d(i)X1X2+e(i)X1X3+f(i)X2X3+g(i)X1^2+h(i)X2^2+i(i)X3^2 i=[1,4]

It consists of four outputs (Y1-Y4), 3 inputs (X1-X3), and (9*4) coeff.

I intend to generate X1,X2,X3=function(Y1,Y2,Y3,Y4).i.e. Equations of X1,X2,X3. We may assume coefficients of your choice to simplify the solution. Please help me with possible guidance.

2

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.