If $p$ : switch $S_1$ is closed, $q$ : switch $S_2$ is closed, $r$ : switch $S_3$ closed, then the symbolic form of the following switching circuit is equivalent to

Switching Circuit:

If $f(x)=\left\{\begin{array}{cc}\frac{1-\cos 4 x}{x^2} & , \text { if } x<0 \\ \frac{a}{\sqrt{x}} & , \text { if } x=0 \\ \frac{(16+\sqrt{x})^{\frac{1}{2}}-4}{16} & , \text { if } x>0\end{array}\right.$

is continuous at $x=0$, then $\mathrm{a}=$

$$ \int \sec ^{\frac{2}{3}} x \cdot \operatorname{cosec}^{\frac{4}{3}} x d x= $$

If the lengths of three vectors $\bar{a}, \bar{b}$ and $\bar{c}$ are $5,12,13$ units respectively, and each one is perpendicular to the sum of the other two, then $|\overline{\mathrm{a}}+\overline{\mathrm{b}}+\overline{\mathrm{c}}|=\ldots \ldots$.