  Recent BESIII data on radiative $J/\psi$ decays from $\sim 10^{10}$ $J/\psi$
samples should significantly advance our understanding of the controversial
nature of $\eta(1405/1475)$. This motivates us to develop a three-body unitary
coupled-channel model for radiative $J/\psi$ decays to three-meson final states
of any partial wave ($J^{PC}$). Basic building blocks of the model are bare
resonance states such as $\eta(1405/1475)$ and $f_1(1420)$, and $\pi K$,
$K\bar{K}$, and $\pi\eta$ two-body interactions that generate resonances such
as $K^*(892)$, $K^*_0(700)$, and $a_0(980)$. This model reasonably fits
$K_SK_S\pi^0$ Dalitz plot pseudo data generated from the BESIII's
$J^{PC}=0^{-+}$ amplitude for $J/\psi\to\gamma K_SK_S\pi^0$. The experimental
branching ratios of $\eta(1405/1475)\to\eta\pi\pi$ and
$\eta(1405/1475)\to\gamma\rho$ relative to that of $\eta(1405/1475)\to
K\bar{K}\pi$ are simultaneously fitted. Our $0^{-+}$ amplitude is analytically
continued to find three poles, two of which correspond to $\eta(1405)$ on
different Riemann sheets of the $K^*\bar{K}$ channel, and the third one for
$\eta(1475)$. This is the first pole determination of $\eta(1405/1475)$ and,
furthermore, the first-ever pole determination from analyzing experimental
Dalitz plot distributions with a manifestly three-body unitary coupled-channel
framework. Process-dependent $\eta\pi\pi$, $\gamma\pi^+\pi^-$, and $\pi\pi\pi$
lineshapes of $J/\psi\to\gamma(0^{-+})\to \gamma(\eta\pi\pi)$,
$\gamma(\gamma\rho)$, and $\gamma(\pi\pi\pi)$ are predicted, and are in
reasonable agreement with data. A triangle singularity is shown to play a
crucial role to cause the large isospin violation of
$J/\psi\to\gamma(\pi\pi\pi)$.
