arcpy.SpatialJoin_analysis(r"C:\Path\To\Points.shp",r"C:\Path\To\Grid.shp",r"C:\Path\To\PointsWithGridData.shp","JOIN_ONE_TO_MANY","KEEP_COMMON")
arcpy.MeanCenter_stats(r"C:\Path\To\PointsWithGridData.shp",r"C:\Path\To\CentralPoints.shp","#","JOIN_FID")
import arcpy inFC=r"C:\Path\To\Grid.shp" outFC=r"C:\Path\To\NewPoints.shp" arcpy.FeatureToPoint_management(inFC,outFC)
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