Let a be the sum of all coefficients in the expansion of (1-2 x+2 x^2)^2023(3-4 x^2+2 x^3)^2024 and b= lim_x → 0((∫_0^x (log (1+t))/(t^2024+1) d…

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