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