Let $$g(x)$$ be a linear function and $$f(x)=\left\{\begin{array}{cl}g(x) & , x \leq 0 \\ \left(\frac{1+x}{2+x}\right)^{\frac{1}{x}} & , x>0\end{array}\right.$$, is continuous at $$x=0$$. If $$f^{\prime}(1)=f(-1)$$, then the value $$g(3)$$ is

The area of the region $$\left\{(x, y): y^2 \leq 4 x, x<4, \frac{x y(x-1)(x-2)}{(x-3)(x-4)}>0, x \neq 3\right\}$$ is

The solution curve of the differential equation $$y \frac{d x}{d y}=x\left(\log _e x-\log _e y+1\right), x>0, y>0$$ passing through the point $$(e, 1)$$ is

Let $$a$$ be the sum of all coefficients in the expansion of $$\left(1-2 x+2 x^2\right)^{2023}\left(3-4 x^2+2 x^3\right)^{2024}$$ and $$b=\lim _\limits{x \rightarrow 0}\left(\frac{\int_0^x \frac{\log (1+t)}{t^{2024}+1} d t}{x^2}\right)$$. If the equation $$c x^2+d x+e=0$$ and $$2 b x^2+a x+4=0$$ have a common root, where $$c, d, e \in \mathbb{R}$$, then $$\mathrm{d}: \mathrm{c}:$$ e equals