solving self consistent equations

I couldn’t attach pic of my self consistent equations in the comment section that I am trying to solve in an earlier message threading

(Elimination of the variables)enter image description here

I am trying to solve for z for given x = 0.057, y = 0.01 self consistently

Code from earlier threading is

y = 1/100;  x = 57/1000;  gE[(n_Integer)?NonNegative] := 1/g[n] - Sqrt[1 + z^2/(Pi*(2*n + 1)*y + (44*x*Pi*g[n])/10)^2];  sce[0] = (2*Pi*1*((Pi*1*(g[0] - 1))/(100*(((Pi*1)/100 + (44*Pi*x)/10)*((Pi*1)/100 + (44*Pi*x*g[0])/10)))))/100 + 6431/10000;  sce[(n_Integer)?Positive] := (2*Pi*1*(((2*n + 1)*Pi*1*(g[n] - 1))/(100*((((2*n + 1)*Pi*1)/100 + (44*Pi*x)/10)*(((2*n + 1)*Pi*1)/100 + (44*Pi*x*g[n])/10)))))/100;  nmax = 3;  nValues = Range[0, nmax];  ee = Flatten[{Total[sce /@ nValues], gE /@ nValues}];  xed0 = GroebnerBasis[ee, {z}, g /@ nValues, MonomialOrder -> EliminationOrder];   NSolve[xed0 == 0, z] 

For nmax = 3 it doesn’t converge and keeps on running and I am aware of @ Daniel Lichtblau comment that here complexity arises as nmax increases.

Question: Is there any other way of solving above self consistent equations?