设\[\begin{aligned}
x&=\sqrt[3]{\cos\dfrac{2\pi}9\cos\dfrac{4\pi}9}+\sqrt[3]{\cos\dfrac{4\pi}9\cos\dfrac{8\pi}9}+\sqrt[3]{\cos\dfrac{8\pi}9\cos\dfrac{2\pi}9},\\
y&=\tan\dfrac{2\pi}{13}\tan\dfrac{5\pi}{13}\tan\dfrac{6\pi}{13}+\tan\dfrac{2\pi}{13}+4\sin\dfrac{6\pi}{13},\\
z&=\dfrac{1}{\sin^2\dfrac{\pi}{14}}+\dfrac{1}{\sin^2\dfrac{3\pi}{14}}+\dfrac{1}{\sin^2\dfrac{5\pi}{14}}
,\end{aligned}\]求 $x+y+z$ 的值．
x&=\sqrt[3]{\cos\dfrac{2\pi}9\cos\dfrac{4\pi}9}+\sqrt[3]{\cos\dfrac{4\pi}9\cos\dfrac{8\pi}9}+\sqrt[3]{\cos\dfrac{8\pi}9\cos\dfrac{2\pi}9},\\
y&=\tan\dfrac{2\pi}{13}\tan\dfrac{5\pi}{13}\tan\dfrac{6\pi}{13}+\tan\dfrac{2\pi}{13}+4\sin\dfrac{6\pi}{13},\\
z&=\dfrac{1}{\sin^2\dfrac{\pi}{14}}+\dfrac{1}{\sin^2\dfrac{3\pi}{14}}+\dfrac{1}{\sin^2\dfrac{5\pi}{14}}
,\end{aligned}\]求 $x+y+z$ 的值．
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