
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
Given a set \( D \) consisting of \( n \) points \( P_{1}, P_{2}, \cdots, P_{n} \) in the plane, where no three points are collinear, each pair of points is connected by a line segment, and all line segments have different lengths. In a triangle, the edge that is neither the longest nor the shortest is called the "middle edge" of the triangle. If each of the three edges of a triangle can be the middle edge of some triangle (not necessarily the same triangle), such a triangle is called a "middle edge triangle" of the set \( D \).

A line \( l \) that does not pass through any point \( P_{i} (i=1,2,\cdots,n) \) divides the set \( D \) into two subsets \( D_{1} \) and \( D_{2} \). For any distribution of these \( n \) points and any choice of line \( l \), there always exists a subset \( D_{k} \) \((k \in \{1,2\})\), such that there exists a middle edge triangle in \( D_{k} \). Determine the minimum value of \( n \).

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