If $A+B+C=\frac{\pi}{2}$, then $\sqrt{2} \cos \left(\frac{\pi}{4}-A\right)$

$$ +\sqrt{2} \cos \left(\frac{\pi}{4}-B\right)+\sqrt{2} \cos \left(\frac{\pi}{4}-C\right)+1= $$

If $\sinh x=\tan A$, then $|\tanh x|=$

$$ \frac{\sinh (x+y)+\sinh (x-y)}{\cosh (x+y)-\cosh (x-y)}= $$

Let $\alpha$ be the period of $3 \sin \frac{\pi x}{3}-\cos \frac{\pi x}{2}+\tan \frac{\pi x}{4}, \beta$ be the period of $\sin ^2\left(\frac{\pi}{7}+\frac{x}{4}\right)-\sin ^2\left(\frac{\pi}{7}-\frac{x}{4}\right)$, and $\gamma$ be the period of $\cos ^4 x+\sin ^4 x$. Then, $\frac{\alpha \gamma}{\beta}=$