
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
An equilateral triangle with side length \( n \) is divided into \( n^2 \) smaller equilateral triangles with side length 1 by lines parallel to its sides. The \( \frac{n(n+1)}{2} \) vertices of the smaller triangles are each labeled with a real number, such that if \( ABC \) and \( BCD \) are small triangles, then the sum of the labels on \( A \) and \( D \) equals the sum of the labels on \( B \) and \( C \). The labels on the vertices of the original triangle are \( a \), \( b \), and \( c \). Determine:

1. The shortest distance between a vertex with the largest label and a vertex with the smallest label.
2. The sum of all the labels.

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