
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
A set \( S \) has a relation \( \rightarrow \) defined on it for pairs of elements from the set \( S \), and it possesses the following properties: 
1) For any two distinct elements \( a, b \in S \), exactly one of the relations \( a \rightarrow b \) or \( b \rightarrow a \) holds.
2) For any three distinct elements \( a, b, c \in S \), if the relations \( a \rightarrow b \) and \( b \rightarrow c \) hold, then the relation \( c \rightarrow a \) also holds.

What is the maximum number of elements that the set \( S \) can contain?

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