
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

There is a deck of 1024 cards, each of which has a set of different digits from 0 to 9 written on them, and all sets are different (`in particular, there is also an empty card`). We call a set of cards complete if each digit from 0 to 9 appears exactly once on them.

Find all natural numbers \( k \) for which there exists a set of \( k \) cards with the following condition: among them, it is impossible to choose a complete set, but adding any card from the deck would violate this condition.

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