Mutate vector by keeping state of the last, newly created, value

Question

I have a data frame like this:

# A tibble: 14 x 1
   inventory
     <int>
 1       3
 2       4
 3       NA
 4       5
 5       NA
 6       NA
 7       NA
 8       NA
 9       10
10       14
11       NA
12       NA
13       NA
14       16

I want to replace NA value with the last available value.

For the row number 3 I can do as follow:

df %>%
  mutate(
    inventory2 = ifelse(
      !is.na(inventory),
      inventory,
      lag(inventory)
    )
  )

But for row 5 to 8, I would like to keep the last value of the newly created vector inventory2. In other words, to reference itself.

I tried the following but it doesn't work

df %>%
  mutate(
    inventory2 = ifelse(
      !is.na(inventory),
      inventory,
      lag(inventory2) # reference to itself doesn't work
    )
  )

for loop is super slow either.

The final output should be:

# A tibble: 14 x 1
   inventory inventory2
       <int>      <int>
 1         3          3
 2         4          4
 3        NA          4
 4         5          5
 5        NA          5
 6        NA          5
 7        NA          5
 8        NA          5
 9        10         10
10        14         14
11        NA         14
12        NA         14
13        NA         14
14        16         16

Is it possible to do this?

Thanks for you help