QAZ likes to play League of Legends and watch league matches as well.
He thinks the double-elimination system of this year's league matches is not fair, so he invented a pro-double-elimination system.
Specifically, in his system:
- There are three teams competing with each other.
- Firstly, two teams are selected with equal probability and compete.
- In each round, the losing team temporarily leaves, and the winning team continues to play against the other of the three teams that did not participate in this round.
- When a team loses two round in total, the elimination competition ends.
- For the sake of simplicity, any two teams have a 50% chance of winning against each other.
- Now please find out the expected number of rounds in this elimination competition.