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r - Find missing month after grouping with dplyr

I have a data frame with two columns that I am grouping by with dplyr, a column of months (as numerics, e.g. 1 through 12), and several columns with statistical data following that (values unimportant). An example:

ID_1   ID_2   month  st1    st2
1      1      1      0.5    0.2
1      1      2      0.7    0.9
1      1      3      1.1    1.7
1      1      4      2.6    0.8
1      1      5      1.8    1.3
1      1      6      2.1    2.2
1      1      7      0.5    0.2
1      1      8      0.7    0.9
1      1      9      1.1    1.7
1      1      10     2.6    0.8
1      1      11     1.8    1.3
1      1      12     2.1    2.2
1      2      1      0.5    0.2
1      2      2      0.7    0.9
1      2      3      1.1    1.7
1      2      4      2.6    0.8
1      2      5      1.8    1.3
1      2      6      2.1    2.2
1      2      7      0.5    0.2
1      2      9      1.1    1.7
1      2      10     2.6    0.8
1      2      11     1.8    1.3
1      2      12     2.1    2.2

For the second grouping (ID_1 = 1 and ID_2 = 2), there is a month missing from the data (month = 8). Is there a way I can find this month and insert a row with the correct ID_1 and ID_2 values, the missing month value, and NA values for the rest of the columns? I've been playing around with this using dplyr functions and can't seem to figure it out, perhaps there is even a non-dplyr solution out there as well.

PS: If it helps, each unique grouping of ID_1 and ID_2 will have no more than 1 month missing.

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Expand grid to make all combos of groups, then merge:

# make reference with all needed rows
ref <- data.frame(expand.grid(unique(df1$ID_1),
                              unique(df1$ID_2),
                              1:12))
colnames(ref) <- colnames(df1)[1:3]

# them merge with all TRUE
res <- merge(df1, ref, all = TRUE)

# to check output, show only month = 8
res[ res$month == 8, ]
#    ID_1 ID_2 month st1 st2
# 8     1    1     8 0.7 0.9
# 20    1    2     8  NA  NA

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