Tranforming dates into Weeks and Fortnights by Months using lubridate

Question

I want to transform dates into weeks and fortnights by months using lubridate R package. My working example is below:

library(tidyverse)
library(lubridate)

dt1 <- 
  tibble(
    Date      = seq(from = ymd("2021-01-01"), to = ymd("2021-12-31"), by = '1 day')
  , Week      = week(Date)
  , Fortnight = ceiling(week(Date)/2)
  , Month     =  month(Date)
    )
dt2 <-
  dt1 %>% 
  group_by(Month) %>% 
  mutate(
    WK = week(Date)
  , FN = ceiling(week(Date)/2)
    ) 

head(dt2, 10)
# A tibble: 10 x 6
# Groups:   Month [1]
   Date        Week Fortnight Month    WK    FN
   <date>     <dbl>     <dbl> <dbl> <dbl> <dbl>
 1 2021-01-01     1         1     1     1     1
 2 2021-01-02     1         1     1     1     1
 3 2021-01-03     1         1     1     1     1
 4 2021-01-04     1         1     1     1     1
 5 2021-01-05     1         1     1     1     1
 6 2021-01-06     1         1     1     1     1
 7 2021-01-07     1         1     1     1     1
 8 2021-01-08     2         1     1     2     1
 9 2021-01-09     2         1     1     2     1
10 2021-01-10     2         1     1     2     1

tail(dt2, 10)
# A tibble: 10 x 6
# Groups:   Month [1]
   Date        Week Fortnight Month    WK    FN
   <date>     <dbl>     <dbl> <dbl> <dbl> <dbl>
 1 2021-12-22    51        26    12    51    26
 2 2021-12-23    51        26    12    51    26
 3 2021-12-24    52        26    12    52    26
 4 2021-12-25    52        26    12    52    26
 5 2021-12-26    52        26    12    52    26
 6 2021-12-27    52        26    12    52    26
 7 2021-12-28    52        26    12    52    26
 8 2021-12-29    52        26    12    52    26
 9 2021-12-30    52        26    12    52    26
10 2021-12-31    53        27    12    53    27

Questions

1. WK (Weeks by Months) and FT (Fortnight by Months) are not as expected.
2. Want to label Weeks as January 01-07, January 08-14, January 15-21, January 22-28, January 29-31, etc.
3. Want to label Fortnights as January 01-14, January 15-28, January 29-31, etc.