Based on $ (10087\pm44)\times10^{6} $$ J/\psi $ events collected with the BESIII detector, the $ J/\psi\rightarrow \gamma K^{0}_{S}K^{0}_{S}\pi^{0} $ and $ J/\psi\rightarrow \gamma \pi^{0}\pi^{0}\eta $ processes are studied. The $ X(2370) $ is observed in both the $ K^{0}_{S}K^{0}_{S}\pi^{0} $ and $ \pi^{0}\pi^{0}\eta $ invariant mass spectra, with statistical significances greater than $ 14\sigma $ and $ 20\sigma $, respectively. By combining measurements from these processes with those from the previously reported $ J/\psi\rightarrow \gamma K^{0}_{S}K^{0}_{S}\eta^{\prime} $ process, the mass and width of the $ X(2370) $ are determined to be $ 2359^{+13}_{-14}\; \text{MeV}/c^{2} $ and $ 170^{+44}_{-29}\; \text{MeV} $, respectively. In addition, the decay $ X(2370)\to a_{0}(980)^{0}\pi^{0} $ with $ a_{0}(980)^{0}\to \pi^{0}\eta $ is observed with a statistical significance exceeding $ 9\sigma $. The properties of the $ X(2370) $—decay pattern similarities to that of $ \eta_{c} $, are consistent with those of a pseudoscalar glueball.