
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 of  $100$  elements. Find the smallest possible  $n$  satisfying the following condition: Given a sequence of  $n$  subsets of  $X$ ,  $A_1,A_2,\ldots,A_n$ , there exists  $1 \leq i < j < k \leq n$  such that  $$ A_i \subseteq A_j \subseteq A_k \text{ or } A_i \supseteq A_j \supseteq A_k. $$  

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