
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
Let  $n,k$  be given natural numbers. Find the smallest possible cardinality of a set  $A$  with the following property: There exist subsets  $A_1,A_2,\ldots,A_n$  of  $A$  such that the union of any  $k$  of them is  $A$ , but the union of any  $k-1$  of them is never  $A$ .

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