  We classify the Deligne-Mumford stacks M compactifying the moduli space of
smooth $n$-pointed curves of genus one under the condition that the points of M
represent Gorenstein curves with distinct markings. This classification
uncovers new moduli spaces $\bar{\mathcal{M}}_{1,n}(Q)$, which we may think of
coming from an enrichment of the notion of level used to define Smyth's
$m$-stable spaces. Finally, we construct a cube complex of Artin stacks
interpolating between the $\bar{\mathcal{M}}_{1,n}(Q)$'s, a multidimensional
analogue of the wall-and-chamber structure seen in the log minimal model
program for $\bar{\mathcal{M}}_g$.
