
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
There are infinitely many cards, each displaying a real number. For each real number \( x \), there is exactly one card displaying the number \( x \). Two players each select a distinct set of 100 cards, \( A \) and \( B \), from these. Establish a rule that satisfies the following conditions to determine the winner between the two players:
1. The winner only depends on the relative order of these 200 cards: If these 200 cards are placed face down in increasing order, with the audience knowing which card belongs to which player but not the actual numbers, the audience can still decide who the winner is;
2. If the elements of the two sets, written in increasing order, are 
   \[
   A = \{a_1, a_2, \ldots, a_{100}\}, \quad B = \{b_1, b_2, \ldots, b_{100}\}
   \]
   such that for all \( i \in \{1,2, \ldots, 100\} \), \( a_i > b_i \), then \( A \) defeats \( B \).
3. If three players each select a distinct set of 100 cards, \( A \), \( B \), and \( C \), and \( A \) defeats \( B \), \( B \) defeats \( C \), then \( A \) defeats \( C \).

Question: How many such rules are there?
(Note: Two rules are considered different if for some sets \( A \) and \( B \), one rule states \( A \) defeats \( B \), while another rule states \( B \) defeats \( A \).)

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