
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]


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For \( n \geq 2 \), an equilateral triangle is divided into \( n^2 \) congruent smaller equilateral triangles. Determine all ways in which real numbers can be assigned to the \(\frac{(n+1)(n+2)}{2}\) vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.

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