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Based on $(10087\pm44)\times10^{6}$ $J/\psi$ events collected with the BESIIIdetector, the process $J/\psi\rightarrow\gamma\pi^{+}\pi^{-}\eta'$ is studiedusing two dominant decay channels of the $\eta'$ meson,$\eta'\rightarrow\gamma\pi^{+}\pi^{-}$ and$\eta'\rightarrow\eta\pi^{+}\pi^{-},\eta\rightarrow\gamma\gamma$. The $X(2600)$is observed with a statistical significance larger than 20$\sigma$ in the$\pi^{+}\pi^{-}\eta'$ invariant mass spectrum, and it has a strong correlationto a structure around 1.5$~\mathrm{GeV}$/{\it c}$^{2}$ in the $\pi^{+}\pi^{-}$invariant mass spectrum. A simultaneous fit on the $\pi^{+}\pi^{-}\eta'$ and$\pi^{+}\pi^{-}$ invariant mass spectra with the two $\eta'$ decay modesindicates that the mass and width of the $X(2600)$ state are $2617.8\pm2.1^{+18.2}_{-1.9}~\mathrm{MeV}$/{\it c}$^{2}$ and$200\pm8^{+20}_{-17}~\mathrm{MeV}$, respectively. The corresponding branchingfractions are measured to be $B(J/\psi\rightarrow\gamma X(2600))\cdotB(X(2600)\rightarrow f_{0}(1500)\eta')\cdotB(f_{0}(1500)\rightarrow\pi^{+}\pi^{-})$ =($3.39\pm0.18^{+0.91}_{-0.66})\times10^{-5}$ and $B(J/\psi\rightarrow\gammaX(2600))\cdot B(X(2600)\rightarrow f_{2}^{'}(1525)\eta')\cdotB(f_{2}^{'}(1525)\rightarrow\pi^{+}\pi^{-})$ =($2.43\pm0.13^{+0.31}_{-1.11})\times10^{-5}$, where the first uncertainties arestatistical, and the second systematic.

Observation of a state $X(2600)$ in the $\pi^{+}\pi^{-}\eta'$ system in the process $J/\psi\rightarrow\gamma\pi^{+}\pi^{-}\eta'$