  The recent BESIII data on $J/\psi\to\gamma(K_SK_S\pi^0)$, which is
significantly more precise than earlier $\eta(1405/1475)$-related data, enables
quantitative discussions on $\eta(1405/1475)$ at the previously unreachable
level. We conduct a three-body unitary coupled-channel analysis of experimental
Monte-Carlo outputs for radiative $J/\psi$ decays via $\eta(1405/1475)$:
$K_SK_S\pi^0$ Dalitz plot distributions from the BESIII, and branching ratios
of $\gamma(\eta\pi^+\pi^-)$ and $\gamma(\gamma\pi^+\pi^-)$ final states
relative to that of $\gamma(K\bar{K}\pi)$. Our model systematically considers
(multi-)loop diagrams and an associated triangle singularity, which is critical
in making excellent predictions on $\eta(1405/1475)\to \pi\pi\pi$ lineshapes
and branching ratios. The $\eta(1405/1475)$ pole locations are revealed for the
first time. Two poles for $\eta(1405)$ are found on different Riemann sheets of
the $K^*\bar{K}$ channel, while one pole for $\eta(1475)$. The
$\eta(1405/1475)$ states are described with two bare states dressed by
continuum states. The lower bare state would be an excited $\eta^\prime$, while
the higher one could be an excited $\eta^{(\prime)}$, hybrid, glueball, or
their mixture. This work presents the first-ever pole determination based on a
manifestly three-body unitary coupled-channel framework applied to experimental
three-body final state distributions (Dalitz plots).
