
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 triangle, let us define the “middle distance” from any line in the plane to the triangle as follows: If the line does not intersect the triangle, the middle distance is the sum of the distances from the vertices to the line. If the line does intersect the triangle, the middle distance is the absolute value of the difference between the sum of the distances from the vertices on one side of the line and the sum of the distances from the vertices on the other side. Where are the lines located for which the middle distance to the triangle is a given value $d$?

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