
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  $S=\{1,2,3,\cdots,100\}$ . Find the maximum value of integer  $k$ , such that there exist  $k$  different nonempty subsets of  $S$  satisfying the condition: for any two of the  $k$  subsets, if their intersection is nonemply, then the minimal element of their intersection is not equal to the maximal element of either of the two subsets.

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