If you won it in 2004, then you may want to file for an automatic extension so that you don't have to file by April 15th (you can get an automatically approved extension of 4 months by filling out the form and you can get until October 15th if you give a reason).

You can then book a European trip or 2 domestic trips and claim the ticket price being offered (print out something from aa.com). If you find a coach fare for $350 and use the miles instead, couldn't you claim that the actual value of the prize was $700?