Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f:[0, \infty) \rightarrow \mathbf{R}$$ be a function defined by $$f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$$. Let $$\mathrm{S}$$ be the set of all points in the interval $$[0,8]$$ at which $$f$$ is not continuous. Then $$\sum_\limits{\text {aes }} a$$ is equal to __________.

Let $$\mathrm{a}>0$$ be a root of the equation $$2 x^2+x-2=0$$. If $$\lim _\limits{x \rightarrow \frac{1}{3}} \frac{16\left(1-\cos \left(2+x-2 x^2\right)\right)}{(1-a x)^2}=\alpha+\beta \sqrt{17}$$, where $$\alpha, \beta \in Z$$, then $$\alpha+\beta$$ is equal to _________.

Let $$f$$ be a differentiable function in the interval $$(0, \infty)$$ such that $$f(1)=1$$ and $$\lim _\limits{t \rightarrow x} \frac{t^2 f(x)-x^2 f(t)}{t-x}=1$$ for each $$x>0$$. Then $$2 f(2)+3 f(3)$$ is equal to _________.

If $$\lim _\limits{x \rightarrow 1} \frac{(5 x+1)^{1 / 3}-(x+5)^{1 / 3}}{(2 x+3)^{1 / 2}-(x+4)^{1 / 2}}=\frac{\mathrm{m} \sqrt{5}}{\mathrm{n}(2 \mathrm{n})^{2 / 3}}$$, where $$\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$$, then $$8 \mathrm{~m}+12 \mathrm{n}$$ is equal to _______.