  Finding a systematic expansion of the spectrum of free superstrings on
AdS${}_5\times $S${}^5$, or equivalently strongly coupled N = 4 SYM in the
planar limit, remains an outstanding challenge. No first principle string
theory methods are readily available, instead the sole tool at our disposal is
the integrability-based Quantum Spectral Curve (QSC). For example, through the
QSC the first five orders in the strong coupling expansion of the conformal
dimension of an infinite family of short operators have been obtained. However,
when using the QSC at strong coupling one must often rely on numerics, and the
existing methods for solving the QSC rapidly lose precision as we approach the
strong coupling regime.
  In this paper, we introduce a new framework that utilises a novel set of QSC
variables with a regular strong coupling expansion. We demonstrate how to use
this approach to construct a new numerical algorithm that remains stable even
at a 't Hooft coupling as large as $10^6$ (or g ~ 100). Employing this
approach, we derive new analytic results for some states in the sl(2) sector
and beyond. We present a new analytic prediction for a coefficient in the
strong coupling expansion of the conformal dimension for the lowest trajectory
at a given twist L. For non-lowest trajectories, we uncover a novel feature of
mixing with operators outside the sl(2) sector, which manifests as a new type
of analytic dependence on the twist.
