The "Pi" Room

Came across this question from a mailing list of pals :

The mighty adventurer came upon the House of "Pi". The house consisted of 9
rooms without windows, all in a row, with a digit of the numerical constant
pi written on the wall of each room:

entrance [3][1][4][1][5][9][2][6][5] exit


There was an entrance door to the first room (the "3" room) and an exit door
at the last room (the second "5" room). There were also doors between each
of the neighboring rooms. Near the entrance there was a sign:

Behold the House of "Pi"

1. You may only pass though the entrance door once.
2. Start with a score of zero.
3. When you enter a room, add the room number to your score.
4. You can only pass though the exit door if your current score is exactly 96.
5. You can't pass though more than 20 doors, including the entrance and exit doors.

By what moves (F and B = Forward and Back, where forward is towards the exit) can the mighty adventurer pass though the House of "Pi"?