HMMT 2018 U21

You are the first lucky player to play in a slightly modified episode of Deal or No Deal! Initially, there are sixteen cases marked 1 through 16. The dollar amounts in the cases are the powers of 2 from $2^{1}=2$ to $2^{16}=65536$, in some random order. The game has eight turns. In each turn, you choose a case and claim it, without opening it. Afterwards, a random remaining case is opened and revealed to you, then removed from the game. At the end of the game, all eight of your cases are revealed and you win all of the money inside them. However, the hosts do not realize you have X-ray vision and can see the amount of money inside each case! What is the expected amount of money you will make, given that you play optimally?