When to keep rolling dice with a doubles wipeout

In a game you repeatedly roll two fair dice, adding the total to your winnings each time. You may stop whenever you like and keep your winnings, but if a roll shows doubles the game ends and you lose everything accumulated. For which current winnings should you roll again?

Roll again while your total is below 35. With current total $x$, one more roll is worth $\tfrac56(x+7)$, since non-double rolls still average 7. This beats stopping exactly when $x<35$; at 35 you are indifferent, and above it you should stop.