\((1)\)函数\(f(x)=\sin x\cos x+\dfrac{\sqrt{3}}{2}\cos {2}x\)的最小正周期是__________．
\((2)\)若\(x\)，\(y\)满足\(\begin{cases}\begin{matrix}x\leqslant 3 \\ x+y\geqslant 2\end{matrix} \\ y\leqslant x\end{cases} \) 则\(x + 2y\)的最大值为_______．
\((3)\)椭圆\( \dfrac{{x}^{2}}{{a}^{2}}+ \dfrac{{y}^{2}}{{b}^{2}}=1\left(a > b > 0\right) \)的左，右焦点分别为\(F_{1}\)，\(F_{2}\)顶点\(B\left( 0,b \right)\)到\(F\)\(2\)的距离为\(4\)，直线\(x=\dfrac{3}{2}a\)上存在点\(P\)，使得\(\triangle {{F}_{2}}P{{F}_{1}}\)为底角是\(30^{\circ}\)的等腰三角形，则此椭圆方程为_____________________．
\((4)\)已知数列\(\{{{a}_{n}}\}\)，令\({{P}_{n}}=\dfrac{1}{n}({{a}_{1}}+2{{a}_{2}}+\cdots {{2}^{n-1}}{{a}_{n}})\left( n\in {{N}_{+}} \right)\)，则称\(\{{{P}_{n}}\}\)为\(\{{{a}_{n}}\}\)的“伴随数列”，若数列\(\left\{ {{a}_{n}} \right\}\)的“伴随数列”\(\{{{P}_{n}}\}\)的通项公式为\({{P}_{n}}={{2}^{n+1}}\left( n\in {{N}_{+}} \right)\)，记数列\(\left\{ {{a}_{n}}-kn \right\}\)的前\(n\)项和为\(S\)\(n\)，若\(S_{n}\leqslant S_{4}\)对任意的正整数\(n\)恒成立，则实数\(k\)取值范围为______________．