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 _______.

If $$\lim _\limits{x \rightarrow 0} \frac{a x^2 e^x-b \log _e(1+x)+c x e^{-x}}{x^2 \sin x}=1$$, then $$16\left(a^2+b^2+c^2\right)$$ is equal to ________.

If the function

$$f(x)= \begin{cases}\frac{1}{|x|}, & |x| \geqslant 2 \\ \mathrm{a} x^2+2 \mathrm{~b}, & |x|<2\end{cases}$$

is differentiable on $$\mathbf{R}$$, then $$48(a+b)$$ is equal to __________.