A fair 6−sided die is rolled repetitively, forming a sequence of values, under the following rules: If any even value is rolled, add it to the current sequence. If a 3 or 5 is rolled, discard the entire sequence and don't add the 3 or 5 to the start of the new sequence. If a 1 is rolled, add the 1 to the current sequence and end the game. Find the expected length of the sequence at the end of the game.
For example if the sequence of rolls is 34265241, we would reset to an empty sequence at the first roll, reset to an empty sequence at the 5, and our final sequence would be 241, which is of length 3.
