
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 \( X = \{A_1, A_2, \cdots, A_n\} \) be a family of distinct 3-element subsets of \( I = \{1, 2, 3, \cdots, 36\} \) such that:
(1) For any \( 1 \leq i < j \leq n \), \( A_i \cap A_j \neq \varnothing \),
(2) \( A_1 \cap A_2 \cap \cdots \cap A_n = \varnothing \).

Find the maximum value of \( n \), and determine the number of such distinct families \( X \) when \( n \) reaches its maximum value.

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