Adding new column for different rows based on the values present in the same row for a different column

I have a dataframe with different combination of factors. each factor is presented in its column (see below)

F1   F2   F3   F4
1     1
1          1  
1                1

I want to add a new column at the end like below

F1   F2   F3   F4  trt
1    1             F1_F2
1          1       F1_F3
1               1  F1_F4

How do I create this column with conditional merging in R. Any advice would be appreciated!

aggregate(ind~row, na.omit(cbind(row = c(row(df)), stack(df))), paste, collapse = "_")
  row   ind
1   1 F1_F2
2   2 F1_F3
3   3 F1_F4

df <- structure(list(F1 = c(1L, 1L, 1L), F2 = c(1L, NA, NA), F3 = c(NA, 
1L, NA), F4 = c(NA, NA, 1L)), class = "data.frame", row.names = c(NA, 
-3L))

This solution utilizes dplyr::mutate in combination with dplyr::across to replace all 1 values with the column name. tidyr::unite() can then be used to create the trt column.

library(tidyverse)

tibble(
  F1 = rep(1, 3),
  F2 = c(1, 0, 0),
  F3 = c(0, 1, 0),
  F4 = c(0, 0, 1),
) |>
  mutate(across(everything(),
                \(x) ifelse(x == 1, cur_column(), NA))) |>
  unite("trt", remove = FALSE, na.rm = TRUE)
#> # A tibble: 3 × 5
#>   trt   F1    F2    F3    F4   
#>   <chr> <chr> <chr> <chr> <chr>
#> 1 F1_F2 F1    F2    <NA>  <NA> 
#> 2 F1_F3 F1    <NA>  F3    <NA> 
#> 3 F1_F4 F1    <NA>  <NA>  F4