Paying for Another Roll

A fair six-sided die pays you its face value in dollars. The first roll is free. After seeing any roll you may either bank that amount and stop, or pay a fee of $\$0.50$ and roll again -- as many times as you like, with the fee due before each new roll and no going back to a face you have declined.

(a) What is the optimal policy, and what is the game worth? (b) For which fees would you hold out for a $6$ only?