  "Resummed-Range Effective Field Theory'' is a consistent nonrelativistic
effective field theory of contact interactions with large scattering length $a$
and an effective range $r_0$ large in magnitude but negative. Its leading order
is non-perturbative. Its observables are universal, i.e.~they depend only on
the dimensionless ratio $\xi:=2r_0/a$, with the overall distance scale set by
$|r_0|$. In the two-body sector, the position of the two shallow $S$-wave poles
in the complex plane is determined by $\xi$. We investigate three identical
bosons at leading order for a two-body system with one bound and one virtual
state ($\xi\le0$), or with two virtual states ($0\le\xi<1$). Such conditions
might, for example, be found in systems of heavy mesons. We find that no
three-body interaction is needed to renormalise (and stabilise) Resummed-Range
EFT at LO. A well-defined ground state exists for
$0.366\ldots\ge\xi\ge-8.72\ldots$. Three-body excitations appear for even
smaller ranges of $\xi$ around the ``quasi-unitarity point'' $\xi=0$
($|r_0|\ll|a|\to\infty$) and obey discrete scaling relations. We explore in
detail the ground state and the lowest three excitations and parametrise their
trajectories as function of $\xi$ and of the binding momentum $\kappa_2^-$ of
the shallowest \twoB state from where three-body and two-body binding energies
are identical to zero three-body binding. As $|r_0|\ll|a|$ becomes
perturbative, this version turns into the ``Short-Range EFT'' which needs a
stabilising three-body interaction and exhibits Efimov's Discrete Scale
Invariance. By interpreting that EFT as a low-energy version of Resummed-Range
EFT, we match spectra to determine Efimov's scale-breaking parameter
$\Lambda_*$ in a renormalisation scheme with a ``hard'' cutoff. Finally, we
compare phase shifts for scattering a boson on the two-boson bound state with
that of the equivalent Efimov system.
