
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
Divide each side of an equilateral triangle $ABC$ with side length 1 into $n$ equal parts. Draw lines through these division points parallel to the other two sides of the triangle, thus dividing the original triangle into smaller triangles. Each vertex of these smaller triangles is called a node, and a real number is placed at each node. Given:
(i) Real numbers $a$, $b$, and $c$ are placed at points $A$, $B$, and $C$, respectively;
(ii) In each rhombus formed by a pair of adjacent smallest triangles, the sum of the numbers placed at opposite vertices is equal.

Find:
(1) The distance $r$ between the node with the maximum number and the node with the minimum number;
(2) The total sum $S$ of the numbers at all nodes.


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