The Envelope You Already Opened

A dealer picks an amount $M$ uniformly from $\{\$100,\, \$200,\, \$400\}$, puts $M$ into one envelope and $2M$ into the other, seals both, and hands you one at random. You open yours and find $\$200$.

(a) Should you swap? (b) Work out the right policy for every amount you might have seen. (c) A colleague argues: "whatever is in your envelope, the other holds either double or half with equal chance, so swapping is worth $\tfrac{1}{2}(2x) + \tfrac{1}{2}(x/2) = 1.25x$ -- always swap, and swap again forever." Where exactly does that break, and how much is your correct policy actually worth?