
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 \( X \) be a set with 100 elements. Find the smallest positive integer \( n \) such that for any sequence \( A_{1}, A_{2}, \cdots, A_{n} \) of subsets of \( X \) of length \( n \), there exist \( 1 \leqslant i < j < k \leqslant n \) satisfying \( A_{i} \subseteq A_{j} \subseteq A_{k} \) or \( A_{i} \supseteq A_{j} \supseteq A_{k} \).

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