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To understand the behavior of the energies as a function of $U$, we first roughly grouped the structures by the behavior of the $E$ vs. $U-J$ curve, which we have indicated by colors in Fig.~\ref{fig:reordering}. The largest group has a slope of roughly $∆E \sim 0.3U$. A second subset has energies that a relatively constant $(∆E \sim const)$. The third group, which appeared to have the lowest energies in the U = 0 run, rapidly increases in energy with $∆E \sim 0.7U$.  In order the rationalize this behavior, we begin by examining the structural motifs associated with each group. Shown in Fig.~\ref{fig:bacoso-structs} are representative structures for each group.