
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
Assume a set of points \( S \) in the plane has the properties: (i) any three points are not collinear; (ii) the distances between any two points are all distinct. For two points \( A \) and \( B \) in \( S \), if there exists \( C \in S \) such that \( |AC| < |AB| < |BC| \), then \( AB \) is called a median edge of \( S \). For three points \( A \), \( B \), and \( C \) in \( S \), if \( AB \), \( AC \), and \( BC \) are all median edges of \( S \), then \( \triangle ABC \) is called a median triangle of \( S \). Find the smallest \( n \) such that any \( n \)-point set \( S \) satisfying properties (i) and (ii) must contain a median triangle.

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