Turns thousands of dummy variables into multinomial variable

Question

I have a dataframe of the following sort:

    a<-c('q','w')
    b<-c(T,T)
    d<-c(F,F)
    .e<-c(T,F)
    .f<-c(F,F)
    .g<-c(F,T)
    h<-c(F,F)
    i<-c(F,T)
    j<-c(T,T)
    
    df<-data.frame(a,b,d,.e,.f,.g,h,i,j)

      a    b     d    .e    .f    .g     h     i    j
    1 q TRUE FALSE  TRUE FALSE FALSE FALSE FALSE TRUE
    2 w TRUE FALSE FALSE FALSE  TRUE FALSE  TRUE TRUE

I want to turn all variables starting with periods at the start into a single multinomial variable called Index such that the second row would have a value 1 for the Index column, the third row would have a value 2, etc. :

df$Index<-c('e','g')

  a    b     d    .e    .f    .g     h     i    j Index
1 q TRUE FALSE  TRUE FALSE FALSE FALSE FALSE TRUE     e
2 w TRUE FALSE FALSE FALSE  TRUE FALSE  TRUE TRUE     g

Although many rows can have a T for any of period-initial variable, each row can be T for only ONE period-initial variable.

If it were just a few items id do an ifelse statement:

df$Index <- ifelse(df$_10000, '10000',...

But there are 12000 of these. The names for all dummy variables begin with underscores, so I feel like there must be a better way. In pseudocode I would say something like:

for every row:
    for every column beginning with '_':
        if value == T:
            assign the name of the column without '_' to a Column 'Index'

Thanks in advance