When crossing through a dangerous cave, you are attacked by a monster, whom you battle and defeat in exponential time with rate m. If you defeat the monster in less than 2/m units of time, you can then safely exit the cave. However, if you take more than 2/m units of time to defeat it, another monster notices you and you then have to defeat it as well, which you do again in independent exponential time with rate m. This same pattern repeats itself until you are able to defeat a monster in less than 2/m units of time.

i) Find the expected time you spend in the cave.

ii) Find the probability that you have to defeat at most 3 monsters before you can exit the cave.