Question 130
Suppose points $F_1, F_2$ are the left and right foci of the ellipse $\frac{x^2}{16} + \frac{y^2}{4} = 1$ respectively, and point $P$ is on line $l$: $x - \sqrt{3}y + 8 + 2\sqrt{3} = 0$. When $\angle F_1PF_2$ reaches the maximum, the value of ratio $\frac{|PF_1|}{|PF_2|}$ is ______.