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