Question: how to find solution minimize the cost function with this method

3*rho1 - 2*rho2 + rho3 - rho4 = -1

4*rho1 +   rho2 - rho3        = 5

original without cost function:

with(Groebner):
K := {y1-(x1^3)*(x2^4),y2-(x2^(1+2))*(w^2),y3-(x1^(1+1))*(w^1),y4-(x2^1)*w,(y1^1000)*(y2^1)*(y3^1)*(y4^100)- x1*x2*w + 1};
G := Basis(K, plex(x1, x2, w, y1, y2, y3, y4));
Reduce((x2^(5+1))*(w^1), G, plex(x1, x2, w, y1, y2, y3, y4));

after have cost function 1000*rho1 + rho2 + rho3 + 100*rho4, i guess to introduce new variable x3

however, the reuslt is not equal to the correct solution (1,3,2,0)

y1-(x1^3)*(x2^4)*(x3^1000)
y2-(x2^(3))*(x3^(3))*(w^2)
y3-(x1^(2))*(x3^(2))*(w^1)
y4-(x2^1)*(x3^(101))*w
x1*x2*x3*w - 1

with(Groebner):
K := {y1-(x1^3)*(x2^4)*(x3^1000),y2-(x2^(1+2))*(x3^(1+2))*(w^2),y3-(x1^(1+1))*(x3^(1+1))*(w^1),y4-(x2^1)*(x3^(100+1))*w,x1*x2*x3*w - 1};
G := Basis(K, plex(x1, x2, w, y1, y2, y3, y4));
Reduce((x2^(5+1))*(w^1), G, plex(x1, x2, w, y1, y2, y3, y4));

subs({rho1=1,rho2=1,rho3=0,rho4=2},3*rho1 - 2*rho2 + rho3 - rho4);
subs({rho1=1,rho2=1,rho3=0,rho4=2},4*rho1 +   rho2 - rho3       );
subs({rho1=1,rho2=1,rho3=0,rho4=2},1000*rho1 + rho2 + rho3 + 100*rho4 );

subs({rho1=1,rho2=3,rho3=2,rho4=0},3*rho1 - 2*rho2 + rho3 - rho4);
subs({rho1=1,rho2=3,rho3=2,rho4=0},4*rho1 +   rho2 - rho3       );
subs({rho1=1,rho2=3,rho3=2,rho4=0},1000*rho1 + rho2 + rho3 + 100*rho4 );