\begin{document}$2.7\times10^{9}$\end{document} \begin{document}$\psi(3686)$\end{document} events collected by the BESIII detector operating at the BEPCII collider, the decay \begin{document}$\psi(3686)\to\gamma\chi_{cJ}, \chi_{cJ}\to\eta\eta\eta^{\prime}$\end{document} is analyzed. The decay modes \begin{document}$\chi_{c1}$\end{document} and \begin{document}$\chi_{c2}\to\eta\eta\eta^{\prime}$\end{document} are observed for the first time, and their corresponding branching fractions are determined to be \begin{document}${\cal{B}}(\chi_{c1}\to\eta\eta\eta^{\prime}) = $\end{document}\begin{document}$ (1.40\, \pm 0.13\, (\text{stat.}) \pm 0.09\, (\text{sys.})) \times 10^{-4}$\end{document} and \begin{document}${\cal{B}}(\chi_{c2}\to\eta\eta\eta^{\prime}) = (4.18\, \pm 0.84\, (\text{stat.}) \pm 0.48\, (\text{sys.})) \times 10^{-5}$\end{document}. An upper limit on the branching fraction of \begin{document}$\chi_{c0}\to\eta\eta\eta^{\prime}$\end{document} is set as \begin{document}$2.59 \times 10^{-5}$\end{document} at a 90% confidence level (CL). A partial wave analysis (PWA) of the decay \begin{document}$\chi_{c1}\to\eta\eta\eta^{\prime}$\end{document} is performed to search for the \begin{document}$1^{-+}$\end{document} exotic state \begin{document}$\eta_1(1855)$\end{document}. The PWA result indicates that the structure in the \begin{document}$\eta\eta^{\prime}$\end{document} mass spectrum is attributed to \begin{document}$f_0(1500)$\end{document}, while in the \begin{document}$\eta\eta$\end{document} mass spectrum, it is attributed to the \begin{document}$0^{++}$\end{document} phase space. The upper limit of \begin{document}${\cal{B}}(\chi_{c1}\to\eta_{1}(1855)\eta) \cdot {\cal{B}}(\eta_{1}(1855)\to\eta\eta^{\prime})< $\end{document}\begin{document}$ 9.79 \times 10^{-5}$\end{document} is set based on the PWA at 90% CL."> Search for <i>η</i><sub>1</sub>(1855) in <i>χ</i><sub><i>cJ</i></sub> → <i>ηηη</i>′ decays -
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