Asymmetric complete resolutions and vanishing of Ext over Gorenstein rings

We construct a class of Gorenstein local rings $R$ which admit minimal complete $R$-free resolutions $\text{bd}C$ such that the sequence $\{{\text{rank}}_R{C}_i\}$ is constant for $i<0$, and grows exponentially for all $i>0$. Over these rings we show that there exist finitely generated $R$-modules $M$ and $N$ such that ${\text{Ext}}_R^i(M,N)=0$ for all $i>0$, but ${\text{Ext}}_R^i(N,M)\mathrm{\ne }0$ for all $i>0$.