Finding Point on Segment that Minimizes Angle with Plane Normal

Question

I'm working on a problem involving geometry in 3D space, and I'm a bit stuck. Here's the problem:

Given:

1. A point $P$ in 3D space.
2. A normal vector $N$ to the plane that contains $P$.
3. A segment defined by points $L_1$ and $L_2$.

I need to determine the point $X$ lying on segment $L_1L_2$ such that the angle between $PX$ and the normal $N$ is minimized.

I've approached the problem by first parameterizing the segment and then trying to express the cosine of the angle between $PX$ and $N$ to later maximize it. However, I'm having difficulty setting up the correct equations and working through the math.

Could anyone provide insights or guidance on how to tackle this problem?

Thank you in advance!