JEE Advanced 2024 Mathematics question on Conic Sections

A normal with slope $\dfrac{1}{\sqrt{6}}$ is drawn from the point $(0,-\alpha)$ to the parabola $x^2 = -4ay$, where $a > 0$. Let L be the line passing through $(0,-\alpha)$ and parallel to the directrix of the parabola. Suppose that L intersects the parabola at two points A and B. Let r denote the length of the latus rectum and s denote the square of the length of the line segment AB. If $r : s = 1 : 16$, then the value of $24a$ is _________.