设向量 $\stackrel{\rightharpoonup }{a}=\left(4\cos \alpha ,\sin \alpha \right),\stackrel{\rightharpoonup }{b}=\left(\sin \beta ,4\cos \beta \right),\stackrel{\rightharpoonup }{c}\left(\cos \beta ,-4\sin \beta \right)$
（1）若 $\stackrel{\rightharpoonup }{a}$与 $\stackrel{\rightharpoonup }{b}-2\stackrel{\rightharpoonup }{c}$垂直，求 $\tan \left(\alpha +\beta \right)$的值；

（2）求 $\left|\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{c}\right|$的最大值;
（3）若 $\tan \alpha tan\beta =16$，求证： $\stackrel{\rightharpoonup }{a}//\stackrel{\rightharpoonup }{b}$.