已知平面向量 $\boldsymbol{e_1},\boldsymbol{e_2}$ 满足 $|2\boldsymbol{e_1}-\boldsymbol{e_2}|\leqslant\sqrt2$，设 $\boldsymbol{a}=\boldsymbol{e_1}+\boldsymbol{e_2},\boldsymbol{b}=3\boldsymbol{e_1}+\boldsymbol{e_2}$，向量 $\boldsymbol{a},\boldsymbol{b}$ 的夹角为 $\theta$，则 $\cos^2\theta$ 的最小值为 ．