35．（2023•北辰区三模）在$\Delta ABC$中，$\overrightarrow{BA}=\overrightarrow{a}$，$\overrightarrow{BC}=\overrightarrow{b}$，若$O$为其重心，试用$\overrightarrow{a}$，$\overrightarrow{b}$表示$\overrightarrow{BO}$为 ___$\frac{1}{3}\overrightarrow{a}+\frac{1}{3}\overrightarrow{b}$___；若$O$为其外心，满足$\frac{AB}{BC}\overrightarrow{BC}\cdot \overrightarrow{BO}+\frac{BC}{AB}\overrightarrow{BA}\cdot \overrightarrow{BO}=2m{(\overrightarrow{BO})}^{2}(m\in R)$，且$\sin A+\sin C=\sqrt{2}$，则$m$的最大值为 [　　]