\((1)\)在边长为\(1\)的正方形\(ABCD\)中,\(2\overrightarrow{AE}=\overrightarrow{EB}\),\(BC\)的中点为\(F\),\(\overrightarrow{{EF}}={2 }\overrightarrow{{FG}}\),则\(\overrightarrow{EG}\cdot \overrightarrow{BD}=\)________.
\((2)\)已知实数\(x\),\(y\)满足约束条件\(\begin{cases} & 3x+y\leqslant \pi \\ & x\geqslant \dfrac{\pi }{6} \\ & y\geqslant 0 \end{cases}\)则\(\sin (x+y)\)的取值范围为________\((\)用区间表示\()\).
\((3)\)某几何体的三视图如图所示,且该几何体的体积是\(\dfrac{3}{2}\),则正视图中的\(x\)的值是________.
\((4)\)已知\(A\)、\(B\)是圆\(C\):\(x^{2}+y^{2}-8x-2y+16=0\)上的两点,点\(P\)在抛物线\(x^{2}=2y\)上,当\(∠APB\)取得最大值时,\(|AB|=\)________.