
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 \( X_{1}, X_{2}, \ldots, X_{100} \) be a sequence of mutually distinct nonempty subsets of a set \( S \). Any two sets \( X_{i} \) and \( X_{i+1} \) are disjoint and their union is not the whole set \( S \), that is, \( X_{i} \cap X_{i+1}=\emptyset \) and \( X_{i} \cup X_{i+1} \neq S \), for all \( i \in \{1, \ldots, 99\} \). Find the smallest possible number of elements in \( S \).

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