
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

Let \( n \geq 4 \) be an integer. Find all functions \( W:\{1, \ldots, n\}^{2} \rightarrow \mathbb{R} \) such that for every partition \( [n]=A \cup B \cup C \) into disjoint sets,
$$
\sum_{a \in A} \sum_{b \in B} \sum_{c \in C} W(a, b) W(b, c)=|A||B||C| .
$$

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