r – Separting alphanumeric string using tidyr separate wider regex

I have the following data,

id <- c("case1", "case19", "case88", "case77")
vec <- c("One_20 (19)",
         "tWo_20 (290)",
         "Three_38 (399)",
         NA)

df <- data.frame(id, vec)

> df
      id            vec
1  case1    One_20 (19)
2 case19   tWo_20 (290)
3 case88 Three_38 (399)
4 case77           

I want to separte the vec vector into two variables, namely: txt and num. I am preferring to use tidyr in this way,

df |> tidyr::separate_wider_regex(vec, 
                                   c(txt = "[A-Za-z]+", num = "\\d+"),
                                   too_few = "align_start")
# A tibble: 4 × 3
  id     txt   num  
     
1 case1  One   NA   
2 case19 tWo   NA   
3 case88 Three NA   
4 case77 NA    NA  

However, it is not what I want. I have the following expection:

      id      txt num
1  case1   One_20  19
2 case19   tWo_20 290
3 case88 Three_38 399
4 case77       NA

I am doing mistakes in the regex part. Any help to correct those mistakes so that I can have the expected table as output?