Barge Problem: Take a chance!

You have made it to the final round of the rebooted 1950’s game show, “Take-A-Chance” and because of your unbelievable luck, you have so far amassed $100,000,000 in winnings! For the final round, the game show wheels out a giant machine consisting of 6 boxes (labeled 1 through 6), each containing $1. The emcee explains that you can “spend” $1 from a given box, say box i, and choose for the game show to either:
      a) place $2 in box i+1, or
      b) interchange the contents of boxes i+1 and i+2.  
You can repeat the above as many times as possible. The emcee next gives you the following choice: you can either quit the game now and keep your $100,000,000 or, you can forfeit these previous winnings, and instead play this last game keeping all the money you generate from its play.
What should you do?
Solutions can be emailed to Professor Jonathan Bloom (bloomjs@lafayette.edu). Submissions are due by 6 a.m. on Saturday, Oct. 12.  

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