
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

For any natural number \( n \), determine the coefficients \( a_{1}, a_{2}, \ldots, a_{n} \) such that the bivariate polynomial

\[ P(x, y)=x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\ldots+a_{n-1} x y^{n-1}+a_{n} y^{n} \]

satisfies the equation

\[ P(u+v, w)+P(v+w, u)+P(w+u, v)=0 \]

for all \( u, v, w \).

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