I have the following data.table

set.seed(1)
DT <- data.table(VAL = sample(c(1, 2, 3), 10, replace = TRUE))
    VAL
 1:   1
 2:   2
 3:   2
 4:   3
 5:   1
 6:   3
 7:   3
 8:   2
 9:   2
10:   1

Now I want to to perform two tasks:

1. Count the occurrences of numbers in VAL.
2. Count within all rows with the same value VAL (first, second, third occurrence)

At the end I want the result

    VAL COUNT IDX
 1:   1     3   1
 2:   2     4   1
 3:   2     4   2
 4:   3     3   1
 5:   1     3   2
 6:   3     3   2
 7:   3     3   3
 8:   2     4   3
 9:   2     4   4
10:   1     3   3

where COUNT defines task 1. and IDX task 2.

I tried to work with which and length using .I:

 dt[, list(COUNT = length(VAL == VAL[.I]), 
             IDX = which(which(VAL == VAL[.I]) == .I))]

but this does not work as .I refers to a vector with the index, so I guess one must use .I[]. Though inside .I[] I again face the problem, that I do not have the row index and I do know (from reading data.table FAQ and following the posts here) that looping through rows should be avoided if possible.

So, what's the data.table way?