4.
\((1)\)矩形\(ABCD\)中,\(AB=4\),\(BC=3\)沿\(AC\)将矩形\(ABCD\)折成一个二面角\(B—AC—D\)为\(120^{\circ}\)的四面体,则四面体\(ABCD\)的外接球的体积是________.
\((2)\)已知\(\overrightarrow{a}=(2\sin 13{}^\circ ,2\sin 77{}^\circ )\),\(|\overrightarrow{a}-\overrightarrow{b}|=1\),\(\left\langle \overrightarrow{a},\overrightarrow{a}-\overrightarrow{b} \right\rangle =\dfrac{\pi }{3}\),则\(|\overrightarrow{a}+\overrightarrow{b}|=\_\_\_\_\_\_\_\_\).
\((3)\)已知函数\(f(x)=\begin{cases} & -{{2}^{|x-1|}}+3,0\leqslant x < 2 \\ & \dfrac{1}{2}f(x-2),x > 2 \end{cases}\),则函数\(g(x)=xf(x)-1\)的零点个数为________.
\((4)\)在数列\(\{a_{n}\}\)及\(\{b_{n}\}\)中,\({{a}_{n+1}}={{a}_{n}}+{{b}_{n}}+\sqrt{a_{n}^{2}+b_{n}^{2}}\),\({{b}_{n+1}}={{a}_{n}}+{{b}_{n}}-\sqrt{a_{n}^{2}+b_{n}^{2}}\),\(a_{1}=1\),\(b_{1}=1.\)设\({{c}_{n}}={{2}^{n}}(\dfrac{1}{{{a}_{n}}}+\dfrac{1}{{{b}_{n}}})\),则数列\(\{c_{n}\}\)的前\(n\)项和为________.