Terminating Sequence expected length

I was preparing for a quant interview and I came across a puzzle on QuantGuide(named Sequence Terminator):
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.
My approach:
Let us define 2 variables $\alpha_n$ and $\beta_n$ as the expected length till the $n^{th}$ iteration given that in the last roll, we continue with the game or the game terminates respectively. But I'm unable to proceed further.
It would be great if you could please help me out.