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