  We show that whenever a closed symplectic manifold admits a Hamiltonian
diffeomorphism with finitely many simple periodic orbits, the manifold has a
spherical homology class of degree two with positive symplectic area and
positive integral of the first Chern class. This theorem encompasses all known
cases of the Conley conjecture (symplectic CY and negative monotone manifolds)
and also some new ones (e.g., weakly exact symplectic manifolds with
non-vanishing first Chern class).
  The proof hinges on a general Lusternik-Schnirelmann type result that, under
some natural additional conditions, the sequence of mean spectral invariants
for the iterations of a Hamiltonian diffeomorphism never stabilizes. We also
show that for the iterations of a Hamiltonian diffeomorphism with finitely many
periodic orbits the sequence of action gaps between the "largest" and the
"smallest" spectral invariants remains bounded and, as consequence, establish
some new cases of the $C^\infty$-generic existence of infinitely many simple
periodic orbits.
