\((1)\)若实数\(x\)，\(y\)满足约束条件\(\begin{cases} y-x\leqslant 1 \\ x+y\leqslant 3 \\ y\geqslant 1 \end{cases},\)则\(z=x+3y\)的最大值为____．

\((2)\)已知等差数列^{\(\left\{{a}_{n}\right\} \)}的前项和为\(S_{n}\)，若\(a_{3}+a_{4}=18-a_{6}-a_{5}\)，则\(S_{8}=\)________．

\((3)\)若向量\(\overset{⇀}{a} \)，\(\overset{⇀}{b} \)夹角为\(\dfrac{\pi }{3}\)，且\(\left| \overset{⇀}{a}\right|=2,\left| \overset{⇀}{b}\right|=1 \)，则\(\overset{⇀}{a} \)与\(\overset{⇀}{a} +2\overset{⇀}{b} \)的夹角为_________ 

\((4)\)已知实数\(a,b\)满足\(\ln \left( b+1 \right)+a-3b=0\)，实数\(c,d\)满足\(2d-c+\sqrt{5}=0\)，则\({{\left( a-c \right)}^{2}}+{{\left( b-d \right)}^{2}}\)的最小值为_________