Count NAs between first and last occured numbers

Here is my toy dataset

df <- tribble(
  ~x, ~y, ~z,
  7,   NA, 4,
  8,   2,  NA,
  NA,  NA, NA,
  NA,  4,  6)

I want to get a dataframe with a number of NAs for each variable only between the first and the last occurrence of numbers in each column and number of NAs between the first occurred number and last row. So, for this example, the desired solution is

desired_df <- tribble(~vars, ~na_count_between_1st_last_num, ~na_count_between_1st_num_last_row,
                       "x",     0,                              2,
                       "y",     1,                              1,
                       "z",     2,                              2)

How can I get the desired output?