Final length of a die sequence with resets

A fair six-sided die is rolled over and over while you maintain a running sequence:

• an even result ($2$, $4$ or $6$) is appended to the sequence;
• a $3$ or a $5$ wipes the sequence clean (the $3$ or $5$ itself is not recorded);
• a $1$ is appended and the game stops.

For instance, the rolls $3,4,2,6,5,2,4,1$ leave the final sequence $2,4,1$ of length $3$. What is the expected length of the sequence when the game ends?

$2$. The final sequence is the run of rolls after the last $3$ or $5$, up to the first $1$. Looking backwards, that run lasts until the first odd number, and each roll in it is odd with probability $\tfrac12$, so its length is geometric with mean $2$. (Simply ignoring $3$s and $5$s and answering $4$ is the classic trap.)