
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
You start 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?

Present the answer in LaTex format: \boxed{Your answer}
user