Question: is it possible to find back possible input parameter in NormalForm

i use a not good example's polynomials to illustrate the idea

u1:=3*a*c+b^2+c;
u2:=7*a^5*b*c^3;
u3:=a+3*b^2;
T := lexdeg([a,b,c],[e1,e2, e3]);
GB := Basis([e1-u1, e2-u2, e3-u3],T);
result := NormalForm(a*b+b*c, GB, T);

now result is to express a*b+b*c in terms of e1, e2, e3 which represent u1, u2, u3.

is it possible to use preimage to find possible u1,u2,u3 if unknown u1,u2,u3 and given known eqx?

How to use preimage to find possible eqx if given known u1,u2,u3 to find eqx in NormalForm(eqx, GB, T)?

what i confused in code below is that if i know it in terms of -1*e1+2*e2+*e3

it already can be used to find eqx, it seems reasonable to put source1list as unknown to find eqx  or find unknown eq1, eq2, eq3 if given known eqx.

with(RegularChains):
with(ConstructibleSetTools):
source1 := PolynomialRing([e1, e2, e3]);
target1 := PolynomialRing([a, b, c]);
source1list := [...];
target1list := [eq1, eq2, eq3];
cs := PolynomialMapPreimage(target1list, source1list, source1, target1);
Info(cs, source1);
[[e1-e2-e3], [1]]
-1*eq1+2*eq2+*eq3;