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How to separate but not split a data frame by differences in value between different rows?

Time：02-01

I have a large data frame of genetic loci, and I'm trying to group it by gene based on similar start-end values. The difference between the end values between rows should not exceed 2000, for example, to be counted as the same gene.

Note

Lines <- "Start End
5000 6000
12000 13000
37000 38000
38500 39000"
DF <- read.table(text = Lines, header = TRUE)

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rdataframe

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