How do you generate the intersections for one Line with the Fill Curve Node?

I am trying to generate the intersections for one line with the Fill Curve node. I have read the answers on Geometrynodes Curve Intersections but, I wanted to come up with an alternative way to make my intersections.

Here are the conditions I gave myself to work with besides using the Fill Curve node,

• Lines may consist of 2 OR more points
• Not all lines are straight lines
• Multiple Intersections can be generated from multiple lines intersecting the one line
• Lines may or may not be cyclic

and below are my node trees so far.

Node Group Used;

My Approach

To generate 2D intersections by the Fill Curve node, I needed all my lines to be Cyclic. Converting my lines into cyclic shapes however, generated extra lines. With extra lines came extra intersections. My issue occurred after I found some elaborate way to delete these extra intersections. I couldn't keep my node tree from deleting existing points from the line when they coincided/overlapped with my extra intersections.

If possible, I would like to incorporate the extra condition of not deleting extra intersections if, they overlap existing points. Then, I could just merge them to get rid of them.

Blender 4.2.3