
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  $1\le k\le n$  be integers. At most how many  $k$ -element subsets can we select from  $\{1,2,\dots,n\}$  such that for any two selected subsets, one of the subsets consists of the  $k$  smallest elements of their union?

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