How to find and merge duplicate points?

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I have two point shapefiles: Red and Blue. Each point shapefile has a z value which represents a radius (i.e. represented as buffers in the example). A duplicate point is defined as a point that lies within Z distance of another point, which can range from 2 - 10 m. In this example there are ten pairs of duplicate points--seven pairs overlap perfectly, while three pairs overlap by falling within another point's Z value.

I would like to keep only one of each duplicate within Z and merge the two shapefiles together. Solutions in ArcGIS, Python, and R would be especially helpful. How can I find and merge duplicate points?





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