In a game, you have a standard six-faced die, starting with the $1 face up. Each turn, you may either roll the die (randomly changing the upface) or 'take' by cashing out the current upface for its dollar amount. The game does not end when you take, and you may take as many times as you want, including multiple times in a row. For example, you could take 100 times on the initial $1 upface for $100 total. Your strategy is to roll until you see a face of at least n for the first time, then 'take' on that face and continue taking as much as you like. Assuming you choose n optimally, what is your expected payout in this game?