\((1)\)已知实数\(x\),\(y\)满足\(\begin{cases} & x+2y-2\geqslant 0 \\ & 2x+y-4\leqslant 0 \\ & y\leqslant x+1 \end{cases}\),且\(m=\dfrac{x+3y+4}{x+1}\),则实数\(m\)的取值范围________.
\((2).\)双曲线\(\dfrac{{{x}^{2}}}{{{a}^{2}}}-\dfrac{{{y}^{2}}}{{{b}^{2}}}=1\)的左右焦点分别为\(F_{1}\),\(F_{2}\),\(P\)是双曲线右支上一点,\(I\)为\(\triangle PF_{1}F_{2}\)的内心,\(Pl\)交\(x\)轴于\(Q\)点\(.\)若\(|F_{1}Q|=|PF_{2}|\),且\(PI︰IQ=2︰1\),则双曲线的离心率\(e\)的值为________。
\((3).\)若平面向量\(\overrightarrow{{{e}_{1}}},\overrightarrow{{{e}_{2}}}\)满足\(|\overrightarrow{{{e}_{1}}}|=|3\overrightarrow{{{e}_{1}}}+\overrightarrow{{{e}_{2}}}|=2\) ,则\(\overrightarrow{{{e}_{1}}}\)在\(\overrightarrow{{{e}_{2}}}\)方向上投影的最大值是________.
\((4).\)观察下列各式;
\(1^{3}=1\).
\(2^{3}=3+5\).
\(3^{3}=7+9+11\).
\(4^{3}=l3+15+17+19\);
\(……\)
若\(m^{3}(m∈N^{*})\)按上述规律展开后\(.\)发现等式右边含有“\(2017\)”这个数,则\(m\)的值为________.