
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
Given \( X_{1}, X_{2}, \cdots, X_{100} \) as a sequence of non-empty subsets of set \( S \), and all are pairwise distinct. For any \( i \in \{1, 2, \cdots, 99\} \), it holds that
\[ X_{i} \cap X_{i+1} = \varnothing \quad \text{and} \quad X_{i} \cup X_{i+1} \neq S . \]

Determine the minimum number of elements in set \( S \).

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