\((1)\)已知\(\tan α,\tan β \)是方程\({{x}^{2}}+6x+7=0\)的两根，则\(\tan (\alpha +\beta ) =\)_______．

\((2)\)已知\(|\overrightarrow{a}|=1\)，\(|\overrightarrow{b}|=2\)，\(\overrightarrow{a}\)与\(\overrightarrow{b}\)的夹角为\(60{}^\circ \)，则\(\overrightarrow{a}+\overrightarrow{b}\)在\(\overrightarrow{a}\)上的投影为．

\((3)\)已知\(\Delta ABC\)的内角\(A,B,C\)所对的边分别为\(a,b,c\)，若\(\sin A=\dfrac{1}{3}\)，\(b=\sqrt{3}\sin B\)，则\(a=\) __________．

\((4)\)如图，半径为\(2\)的扇形的圆心角为\(120{}^\circ ,M,N\)分别为半径\(OP,OQ\)的中点，\(A\)为弧\(\overset{{}}{{PQ}}\,\)上任意一点，则\(\overrightarrow{AM}\cdot \overrightarrow{AN}\)的取值范围是．