  We produce a model where every supercompact cardinal is
$C^{(1)}$-supercompact with inaccessible targets. This is a significant
improvement of the main identity-crises configuration obtained in \cite{HMP}
and provides a definitive answer to a question of Bagaria \cite[p.19]{Bag}.
This configuration is a consequence of a new axiom we introduce -- called
$\mathcal{A}$ -- which is showed to be compatible with Woodin's $I_0$
cardinals. We also answer a question of V. Gitman and G. Goldberg on the
relationship between supercompactness and cardinal-preserving extendibility. As
an incidental result, we prove a theorem suggesting that supercompactness is
the strongest large-cardinal notion preserved by Radin forcing.
