Find closest line(s) to points, get position on line(s)

[KeithD1]

Given a feature class containing point locations another containing polylines, I would like to a) determine which line(s) are within a specified distance of each point, and b) what is the “position” along the line(s) which comes closest to each point. 

Working in ArcMap 10.4.1 (upgrading to 10.7 soon) with an Advanced license. Will be coding the solution using ArcPy. 

Input Example:

(line 1001 and 1002 are roughly 100 feet apart)

Desired Output:

Thanks for any advice!

[KeithD1]

Maybe if I ask the question a different way?

I'm trying to build something similar to the Linear Referencing tool "Locate Features along Routes". 

Locating features along routes—Help | ArcGIS for Desktop