
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 \( S = \{1, 2, \cdots, 100\} \). Find the largest integer \( k \) such that \( S \) has \( k \) different non-empty subsets with the property that for any two different subsets among these \( k \) subsets, if their intersection is non-empty, then the smallest element in their intersection is not the same as the largest element in either of these two subsets.

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