Let a differentiable function $f$ satisfy the equation $\int_0^{36} f\left(\frac{t x}{36}\right) d t=4 \alpha f(x)$. If $y=f(x)$ is a standard parabola passing through the points $(2,1)$ and $(-4, \beta)$, then $\beta^\alpha$ is equal to $\_\_\_\_$ .

The number of elements in the set $\mathrm{S}=\left\{x: x \in[0,100]\right.$ and $\left.\int\limits_0^x t^2 \sin (x-t) \mathrm{d} t=x^2\right\}$ is $\_\_\_\_$

Let [.] be the greatest integer function. If $\alpha=\int\limits_0^{64}\left(x^{1 / 3}-\left[x^{1 / 3}\right]\right) \mathrm{d} x$, then $\frac{1}{\pi} \int\limits_0^{\alpha \pi}\left(\frac{\sin ^2 \theta}{\sin ^6 \theta+\cos ^6 \theta}\right) \mathrm{d} \theta$ is equal to $\_\_\_\_$ .

Let $[\cdot]$ denote the greatest integer function and $f(x) = \lim\limits_{n \to \infty} \frac{1}{n^{3}} \sum\limits_{k=1}^n \left[ \frac{k^2}{3^x} \right]$. Then $12 \sum\limits_{j=1}^{\infty} f(i)$ is equal to ________.