
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
From a deck of 32 cards which includes three colors (red, yellow, and blue) with each color having 10 cards numbered from $1$ to $10$, plus an additional two cards (a small joker and a big joker) both numbered $0$, a subset of cards is selected. The score for each card is calculated as $2^{k}$, where $k$ is the number on the card. If the sum of these scores equals $2004$, the subset is called a "good" hand. How many "good" hands are there?

(2004 National Girls' Olympiad problem)

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