Multivariate Polynomial division in Macaulay2

Question

I'm trying to perform multivariate polynomial division in macaulay2 where remainder is either 0 or a linear combination of leading terms of the divisors.

One such example is shown in the picture below I have tried doing the same with the // operator by putting divisors in a matrix but that seems to give a different output as shown

CODE :

R=QQ[x,y,z,MonomialOrder=>Lex];
g1=x*y-1;
g2=y^2-1;
g3=x^2*y+y^2*x+y^2;
I= ideal((g2),(g1))
(g3)//gens I
g3% gens I

OUTPUT:

Please help me spot the problem here thank you!