arcpy.FeatureVerticesToPoints_management("ORIGINAL","D:/Scratch/ENDS.shp", "BOTH_ENDS")
arcpy.AddField_management("ENDS","WHAT", "SHORT")
arcpy.CalculateField_management("ENDS", "WHAT", "!FID! % 2", expression_type="PYTHON_9.3")
arcpy.AddField_management("ENDS", "UNIQUE","TEXT", field_length="10")
arcpy.CalculateField_management("ENDS", "UNIQUE", expression="'%s %s' %( !LINE_ID!, !WHAT!)", expression_type="PYTHON_9.3")
arcpy.Near_analysis("ENDS","NODE", search_radius="", location="LOCATION")
arcpy.TableToTable_conversion(in_rows="ENDS", out_path="D:/Scratch", out_name="xy.dbf")

arcpy.Merge_management("DOUBLES;ENDS","D:/Scratch/MERGED.shp")
arcpy.PointsToLine_management("MERGED", "D:/Scratch/connections.shp", Line_Field="UNIQUE")