Another way of thinking of it is that, when someone says, "The boxes are already there, so your decision cannot affect what's in them," he is wrong. It has been assumed that your decision does affect what's in them, so the fact that you cannot imagine how that is possible is wholly irrelevant.

Your decision doesn't affect what's in the boxes, but your decision procedure does, and that already exists when the question's being assigned. It may or may not be possible to derive your decision from the decision procedure you're using in the general case -- I haven't actually done the reduction, but at first glance it looks cognate to some problems that I know are undecidable -- but it's clearly possible in some cases, and it's at least not completely absurd to imagine an Omega with a very high success rate.

As best I can tell, most of the confusion here comes from a conception of free will that decouples the decision from the procedure leading to it.