Create timestamp columns based on multiple filtering conditions (R, dplyr)

I have a dataset, df

 Read      Box       ID      Time                             Subject 
 T         out               10/1/2019 9:00:01 AM
 T         out               10/1/2019 9:00:02 AM             Re:
 T         out               10/1/2019 9:00:03 AM             Re:
 T         out               10/1/2019 9:02:59 AM             Re:
 T         out               10/1/2019 9:03:00 AM
 F                           10/1/2019 9:05:00 AM
 T         out               10/1/2019 9:06:00 AM             Fwd:
 T         out               10/1/2019 9:06:02 AM             Fwd:
 T         in                10/1/2019 9:07:00 AM
 T         in                10/1/2019 9:07:02 AM
 T         out               10/1/2019 9:07:04 AM
 T         out               10/1/2019 9:07:05 AM             Fw:
 T         out               10/1/2019 9:07:06 AM             Fw:
           hello             10/1/2019 9:07:08 AM

Based on certain conditions within this dataset, I would like to create a startime column and an endtime column.

I would like to create a 'starttime' when the following occurs: If the Subject columns first word begins with RE:, re, FWD, or FW (in a consecutive manner), Read == "T", Box == "out" and ID == ""

When the first instance of this condition occurs, a starttime will be generated. For example for this dataset, the starttime will be 10/1/2019 9:00:02 AM since this is where we see the desired conditions occurs first (Subject is either FW:, RE: or FWD, Read = T, Box = out and ID = "" ) However, the moment when anyone of these conditions is not true, an endtime will be created. So the first endtime would occur right before row 4, where the time is 10/1/2019 9:02:59 AM. My ultimate goal is to then create a duration column for this.

This is my desired output when including RE, Fwd, and Fw

  starttime                    endtime                     duration

  10/1/2019 9:00:02 AM        10/1/2019 9:02:59 AM         177 secs
  10/1/2019 9:06:00 AM        10/1/2019 9:06:02 AM         2 secs
  10/1/2019 9:07:05 AM        10/1/2019 9:07:06 AM         1 secs

Furthermore, how would I specify in a separate code to create the start and endtimes for these conditions: Read = T, Box = out, ID = "" and when the the first word in the subject column does not contain Re, Fwd, or Fw?

 Read      Box       ID      Time                             Subject 
 T         out               10/1/2019 9:00:01 AM
 T         out               10/1/2019 9:00:02 AM             Re:
 T         out               10/1/2019 9:00:03 AM             Re:
 T         out               10/1/2019 9:02:59 AM             Re:
 T         out               10/1/2019 9:03:00 AM
 F                           10/1/2019 9:05:00 AM
 T         out               10/1/2019 9:06:00 AM             Fwd:
 T         out               10/1/2019 9:06:02 AM             Fwd:
 T         in                10/1/2019 9:07:00 AM
 T         in                10/1/2019 9:07:02 AM
 T         out               10/1/2019 9:07:04 AM
 T         out               10/1/2019 9:07:05 AM             Fw:
 T         out               10/1/2019 9:07:06 AM             Fw:
           hello             10/1/2019 9:07:08 AM

This is my desired output when excluding RE, Fwd, and Fw

  starttime                    endtime                     duration

  10/1/2019 9:00:01 AM        10/1/2019 9:00:01 AM         0 secs
  10/1/2019 9:03:00 AM        10/1/2019 9:03:00 AM         0 secs
  10/1/2019 9:07:04 AM        10/1/2019 9:07:04 AM         0 secs

dput:

 structure(list(Read = structure(c(3L, 3L, 3L, 3L, 3L, 2L, 3L, 
3L, 3L, 3L, 4L, 4L, 3L, 1L), .Label = c("", "F", "T", "T "), class = "factor"), 
Box = structure(c(3L, 3L, 3L, 3L, 3L, 1L, 3L, 3L, 2L, 2L, 
3L, 3L, 3L, 1L), .Label = c("", "in", "out"), class = "factor"), 
ID = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 2L), .Label = c("", "hello"), class = "factor"), 
Time = structure(1:14, .Label = c("10/1/2019 9:00:01 AM", 
"10/1/2019 9:00:02 AM", "10/1/2019 9:00:03 AM", "10/1/2019 9:02:59 AM", 
"10/1/2019 9:03:00 AM", "10/1/2019 9:05:00 AM", "10/1/2019 9:06:00 AM", 
"10/1/2019 9:06:02 AM", "10/1/2019 9:07:00 AM", "10/1/2019 9:07:02 AM", 
"10/1/2019 9:07:04 AM", "10/1/2019 9:07:05 AM", "10/1/2019 9:07:06 AM", 
"10/1/2019 9:07:08 AM"), class = "factor"), Subject = structure(c(1L, 
4L, 4L, 4L, 1L, 1L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 1L), .Label = c("", 
"Fw:", "Fwd:", "Re:"), class = "factor")), class = "data.frame", row.names = c(NA, 
-14L))

The code that was suggested works, I just would like to also incorporate the Subject column conditions as well:
Where Subject == FW, FWD, RE (ignore the upper/lowercase) and Where Subject is not equal to FW, FWD, Re (ignore upper/lowercase)

library(dplyr)

df %>%
mutate(Time = lubridate::mdy_hms(Time), 
cond = Read == "T"& Box == "out"& ID == ""& Subject == "FW"& Subject  == "FWD"& Subject == "RE" (ignore.case = TRUE)
grp = cumsum(!cond)) %>%
filter(cond) %>%
group_by(grp) %>%
summarise(starttime = first(Time), 
endtime = last(Time), 
duration = difftime(endtime, starttime, units = "secs")) %>%
select(-grp)

library(dplyr)

df %>%
mutate(Time = lubridate::mdy_hms(Time), 
cond = Read == "T"& Box == "out"& ID == ""& Subject! == "FW"& Subject! == "FWD"& Subject! == "RE" (ignore.case = TRUE)
grp = cumsum(!cond)) %>%
filter(cond) %>%
group_by(grp) %>%
summarise(starttime = first(Time), 
endtime = last(Time), 
duration = difftime(endtime, starttime, units = "secs")) %>%
select(-grp)