Consider statements $\mathrm{p}: \mathrm{S}_1$ is closed; $\mathrm{q}: \mathrm{S}_2$ is closed; $\mathrm{r}: \mathrm{S}_3$ is closed. The simplified equivalent circuit diagram and its logical statement for the switching circuit is respectively.

In a triangle $A B C$, with usual notations, $\cot \left(\frac{A+B}{2}\right) \cdot \tan \left(\frac{A-B}{2}\right)=$

In a triangle ABC , with usual notations, $(\mathrm{a}+\mathrm{b}+\mathrm{c})(\mathrm{a}+\mathrm{b}-\mathrm{c})=3 \mathrm{ab}$, then $\angle \mathrm{C}=$

If $\bar{a}, \bar{b}, \bar{c}$ are three coplanar vectors such that $|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2, \overline{\mathrm{~b}} \cdot \overline{\mathrm{c}}=8$, the angle between $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}}$ is $45^{\circ}$, then $|\overline{\mathrm{a}} \times(\overline{\mathrm{b}} \times \overline{\mathrm{c}})|=$