
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
On a plane, there is a set $D$ of $\mathrm{n}$ points $\mathrm{P}_{1}, \mathrm{P}_{2}, \ldots, \mathrm{P}_{\mathrm{n}}$ such that any three points are not collinear. A line segment is drawn between any two points, and the lengths of these line segments are all distinct. In a triangle, the side whose length is neither the longest nor the shortest is called the "middle side" of that triangle. If all three sides of a triangle qualify as "middle sides" (according to the definition of the middle side), then that triangle is referred to as a "middle side triangle" in the set $D$. A straight line $\mathrm{l}$ divides the set $D$ into two subsets $D_{1}$ and $D_{2}$. No matter how the $\mathrm{n}$ points are distributed and no matter how $\mathrm{l}$ is chosen, there always exists a subset $\mathrm{D_k}$ ($\mathrm{k} \in \{1, 2\}$) containing a middle side triangle. Find the minimum value of $\mathrm{n}$.

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