How to compute the $G$-theory of this simplicial toric surface?

Question

Let $k$ be an algebraically closed field of characteristic zero. Let $\sigma_0$ be the cone in $\mathbb{R}^2$ generated by $e_1,e_2$.And let $\sigma_1$ be the cone in $\mathbb{R}^2$ generated by $e_2,-e_1-2e_2$. Let $X$ be the toric variety over the field $k$ defined by the fan built from the two cones $\sigma_0$,$\sigma_1$.I am trying to compute the $G$-theory of $X$. I showed that left multiplication by the matrix $\begin{pmatrix} -1 & 1\\ 0 & 1 \end{pmatrix}$ Maps the cone ${\sigma_1}^{\vee}$ isomorphically onto the cone in $\mathbb{R}^2$ generated by $e_1,3e_1+e_2$. Hence, the $G$-theory of the affine toric variety $U_{\sigma_1}$ agrees with the $G$-theory of $Spec(k[x,x^3y])$. Previously, I computed that $G_n(k[x,x^3y])\cong G_n(k)$ for all non-negative integers $n$. By setting $U=U_{\sigma_0},V=U_{\sigma_1}$, we have that $\{U,V\}$ is an affine open cover of the noetherian scheme $X$, and the intersection $U\cap V$ is the affine toric variety $U_{\sigma_0\cap\sigma_1}$. I computed that $U_{\sigma_0}\cap U_{\sigma_1}$ is isomorphic to Spec$(k[x,x^{-1},y])$. I am trying to use the Mayer-Vietoris sequence of this affine open cover of $X$ to compute the $G$-theory of $X$. Since I have the exact sequence $0\rightarrow ker(\Delta)\rightarrow G_n(X)\rightarrow im(\Delta)\rightarrow 0$ for each $n$,I was hoping that such sequences split. I am also thinking that the boundary map $\partial:G_{n+1}(U\cap V)\rightarrow G_n(X)$ might be computed as $\partial(x\cup\alpha)=\alpha\partial(x)=\alpha$ for every $\alpha\in G_{n}(k[x,x^{-1},y])$,so that $ker(\Delta)=im(\partial)$ , which is equal to $G_{n}(k[x,x^{-1},y])$. In this case of $n=0$, I showed that the boundary image of $x$ in $G_0(Z)$ is 1, where $Z$ is the complement of $U$ in $X$, which is $\mathbb{A}^1_k$ in this case.And I computed that the proper pushforward in $G_0$ by the inclusion of $Z$ into $X$ maps the class $[k[x]]\in G_0(Z)$ to the class $[i_*(O_Z)]\in G_0(X)$, so that the image of the boundary map for the Mayer-Vietoris sequence for $X$ is the cyclic group generated by the class $[i_*(O_Z)]\in G_0(X)$. But I don’t know how to compute the order of the element $[i_*(O_Z)]\in G_0(X)$. And for the case $n>0$, I am stuck. Any help is very much appreciated.