For a real number $\alpha$, let $[\alpha]$ denote the greatest integer less than or equal to $\alpha$. For a finite set $S$, let $|S|$ denote the number of elements in the set $S$.

Consider the functions $f:(-3,3) \rightarrow(-\infty, \infty)$ and $g:(-3,3) \rightarrow(-\infty, \infty)$ defined by

$$ f(x)=\left[x^3\right] \log _e\left(1+\sin ^2(\pi(x-[x]))\right) $$

and

$$ g(x)=x^3 \sin ^2\left(\pi \log _e(1+x-[x])\right) . $$

Let

$$ A=\{x \in(-3,3): f \text { is discontinuous at } x\} $$

and

$$ B=\{x \in(-3,3): g \text { is discontinuous at } x\} . $$

Then the value of $|A|+2|B|-|A \cap B|$ is $\_\_\_\_$ .

Consider the function $f:\left(-\frac{\pi}{2},\frac{\pi}{2}\right) \to (-\infty, \infty)$ defined by

$$f(x) = (|x| + |x-1|) \sin x + \left[ x \sin x \right],$$

where $\left[ x \sin x \right]$ is the greatest integer less than or equal to $x \sin x$.

Let $\alpha$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT continuous, and let $\beta$ be the total number of points in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ at which $f$ is NOT differentiable.

Then the value of $\alpha + \beta$ is ____________.

Let α and β be the real numbers such that

$ \lim\limits_{x \to 0} \frac{1}{x^3} \left( \frac{\alpha}{2} \int\limits_0^x \frac{1}{1-t^2} \, dt + \beta x \cos x \right) = 2. $

Then the value of α + β is ___________.