
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 \( S_{1}, S_{2}, \ldots, S_{100} \) be finite sets of integers whose intersection is not empty. For each non-empty \( T \subseteq \{ S_{1}, S_{2}, \ldots, S_{100} \} \), the size of the intersection of the sets in \( T \) is a multiple of \( |T| \). What is the smallest possible number of elements which are in at least 50 sets?

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