4．
\((1)\)已知向量\( \overrightarrow{a}=(2,-1), \overrightarrow{b}=(1,3) \)，且\(\overrightarrow{a}\bot (\overrightarrow{a}+m\overrightarrow{b})\)，则\(m=\)__________．
\((2)\)已知点\(P\left( \sin \dfrac{3}{4}\pi ,\cos \dfrac{3}{4}\pi \right)\)落在角\(\theta \)的终边上，且\(\theta \in \left[ 0,2\pi \right)\)，则\(\tan \left( \theta +\dfrac{\pi }{3} \right)\)的值为___________．
\((3)\)已知三棱锥\(S-ABC\)的所有顶点都在以\(O\)为球心的球面上，\(\Delta ABC\)是边长为\(1\)的正三角形，\(SC\)为球\(O\)的直径，若三棱锥\(S-ABC\)的体积为\(\dfrac{\sqrt{11}}{6}\)，则球\(O\)的表面积为___________\(.\)
\((4)\)已知\({{F}_{1}},{{F}_{2}}\)为双曲线\(\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\left( a > 0,b > 0 \right)\)的左、右焦点，\(O\)为坐标原点，点\(P\)在双曲线的左支上，点\(M\)在直线\(x=\dfrac{{{a}^{2}}}{c}\left( c=\sqrt{{{a}^{2}}+{{b}^{2}}} \right)\)上，且满足\(\overrightarrow{{{F}_{1}}O}=\overrightarrow{PM},\overrightarrow{OP}=\lambda \left( \dfrac{\overrightarrow{O{{F}_{1}}}}{\overrightarrow{\left| O{{F}_{1}} \right|}}+\dfrac{\overrightarrow{OM}}{\overrightarrow{\left| OM \right|}} \right)\left( \lambda > 0 \right)\)，则该双曲线的离心率为__________．