GAIC Math 2024 Problem 105

Given that the line $l$ intersects two parabolas $\Gamma_{1}: y^{2}=2px (p>0)$ and $\Gamma_{2}: y^{2}=4px$ at four distinct points $A\left(x_{1}, y_{1}\right)$, $B\left(x_{2}, y_{2}\right)$, $D\left(x_{3}, y_{3}\right)$, and $E\left(x_{4}, y_{4}\right)$, where $y_{4}<y_{2}<y_{1}<y_{3}$. Let $l$ intersect the $x$-axis at point $M$. Given that $AD = 6 BE$, then the value of $\frac{AM}{ME}$ is _____.