
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
We have constructed a rhombus by attaching two equal equilateral triangles. By putting  $n-1$  points on all 3 sides of each triangle we have divided the sides to  $n$  equal segments. By drawing line segements between correspounding points on each side of the triangles we have divided the rhombus into  $2n^2$  equal triangles.
We write the numbers  $1,2,\dots,2n^2$  on these triangles in a way no number appears twice. On the common segment of each two triangles we write the positive difference of the numbers written on those triangles. Find the maximum sum of all numbers written on the segments.

(25 points)
*Proposed by Amirali Moinfar*

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