\((1)\)已知变量\(x,y\)满足\(\begin{cases}x-4y+3⩽0 \\ \begin{matrix}x+y-4\leqslant 0 \\ x\geqslant 1\end{matrix}\end{cases} \)，\( \dfrac{{x}^{2}+{y}^{2}}{xy} \)的取值范围为 ．
\((2)\)若向量\(\mathbf{a}=\left( \cos \theta ,\cos \left( \theta +\dfrac{3\pi }{2} \right) \right),\mathbf{b}=\left( -1,2 \right)\)共线，则\({{\sin }^{4}}\theta +{{\cos }^{4}}\theta \)的值为 ．
\((3)\)已知\(p:\exists {{x}_{0}}\in \mathbf{R},a\sin {{x}_{0}}+\cos {{x}_{0}}=-2\)，\(q:f(x)=x-\dfrac{3}{4}a\ln x+\dfrac{3-a}{x}\)在\(\left[ 1,2 \right]\)上为减函数，则\(p\)是\(q\)的 条件．
\((4)\)观察下列等式：\(\dfrac{3}{1\times 2}\times \dfrac{1}{2}=1-\dfrac{1}{{{2}^{2}}},\begin{matrix} {} & {} \\ \end{matrix}\dfrac{3}{1\times 2}\times \dfrac{1}{2}+\dfrac{4}{2\times 3}\times \dfrac{1}{{{2}^{2}}}=1-\dfrac{1}{3\times {{2}^{2}}},\dfrac{3}{1\times 2}\times \dfrac{1}{2}+\dfrac{4}{2\times 3}\times \dfrac{1}{{{2}^{2}}}+\dfrac{5}{3\times 4}\times \dfrac{1}{{{2}^{3}}}=1-\dfrac{1}{4\times {{2}^{3}}},\cdot \cdot \cdot \cdot \cdot \cdot \)
由此推出则第\(n\)个等式为
\((5)\)已知函数\(f(x)=\begin{cases} & \left| {{\log }_{2}}(x+2) \right|-m,x\in \left( -2,0 \right) \\ & \cos \left( \dfrac{\pi }{4}x \right)-m,x\in \left[ 0,8 \right] \\ \end{cases}\)有且仅有\(4\)个零点\({{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\)，且\({{x}_{1}} < {{x}_{2}} < {{x}_{3}} < {{x}_{4}}\)，则\(\dfrac{({{x}_{3}}-2)({{x}_{4}}-2)}{{{x}_{1}}{{x}_{2}}+2({{x}_{1}}+{{x}_{2}})+5}\)的取值范围是 ．