Let $f(x)=\left\{\begin{array}{cc}e^{x-1} & , x<0 \\ x^2-5 x+6 & , x \geq 0\end{array}\right.$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and is not differentiable are $\alpha$ and $\beta$ respectively, then $\alpha+\beta$ is equal to $\_\_\_\_$

Let $\mathrm{A}, \mathrm{B}$ be points on the two half-lines $x-\sqrt{3}|y|=\alpha, \alpha>0$ at a distance of $\alpha$ from their point of intersection $P$. The line segment $A B$ meets the angle bisector of the given half-lines at the point $Q$. If $P Q=\frac{9}{2}$ and $R$ is the radius of the circumcircle of $\triangle \mathrm{PAB}$, then $\frac{\alpha^2}{R}$ is equal to $\_\_\_\_$

Let $\mathrm{A}, \mathrm{B}$ and C be the vertices of a variable right angled triangle inscribed in the parabola $y^2=16 x$. Let the vertex $B$ containing the right angle be $(4,8)$ and the locus of the centroid of $\triangle A B C$ be a conic $C_0$. Then three times the length of latus rectum of $\mathrm{C}_0$ is $\_\_\_\_$

Let $f$ be a twice differentiable function such that

$$ f(x)=\int_0^x \tan (t-x) d t-\int_0^x f(t) \tan t d t, x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) $$

Then $f^{\prime \prime}\left(\frac{\pi}{6}\right)+12 f^{\prime}\left(-\frac{\pi}{6}\right)+f\left(\frac{\pi}{6}\right)$ is equal to _______.