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r - Increment by one to each duplicate value

I am trying to find a proper way, in R, to find duplicated values, and add the value 1 to each subsequent duplicated value grouped by id. For example:

data = data.table(id = c('1','1','1','1','1','2','2','2'),
                  value = c(95,100,101,101,101,20,35,38))

data$new_value <- ifelse(data[ , data$value] == lag(data$value,1),
                         lag(data$value, 1) + 1 ,data$value)
data$desired_value <- c(95,100,101,102,103,20,35,38)

Produces:

   id value new_value desired_value
1:  1    95        NA            95
2:  1   100       100           100
3:  1   101       101           101 # first 101 in id 1: add 0
4:  1   101       102           102 # second 101 in id 1: add 1
5:  1   101       102           103 # third 101 in id 1: add 2
6:  2    20        20            20
7:  2    35        35            35
8:  2    38        38            38

I tried doing this with ifelse, but it doesn't work recursively so it only applies to the following row, and not any subsequent rows. Also the lag function results in me losing the first value in value.

I've seen examples with character variables with make.names or make.unique, but haven't been able to find a solution for a duplicated numeric value.

Background: I am doing a survival analysis and I am finding that with my data there are stop times that are the same, so I need to make it unique by adding a 1 (stop times are in seconds).

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Here's an attempt. You're essentially grouping by id and value and adding 0:(length(value)-1). So:

data[, onemore := value + (0:(.N-1)), by=.(id, value)]

#   id value new_value desired_value onemore
#1:  1    95        96            95      95
#2:  1   100       101           100     100
#3:  1   101       102           101     101
#4:  1   101       102           102     102
#5:  1   101       102           103     103
#6:  2    20        21            20      20
#7:  2    35        36            35      35
#8:  2    38        39            38      38

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