MathNet 2005 — Part 03 Question 170

You start out with a big pile of $3^{2004}$ cards, with the numbers $1,2,3, \ldots, 3^{2004}$ written on them. You arrange the cards into groups of three any way you like; from each group, you keep the card with the largest number and discard the other two. You now again arrange these $3^{2003}$ remaining cards into groups of three any way you like, and in each group, keep the card with the smallest number and discard the other two. You now have $3^{2002}$ cards, and you again arrange these into groups of three and keep the largest number in each group. You proceed in this manner, alternating between keeping the largest number and keeping the smallest number in each group, until you have just one card left.
How many different values are possible for the number on this final card?