
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
Given integers \( n \geq 1 \) and \( n \geq 3 \), determine the two-variable polynomials \( P \) that satisfy the following properties:

a) For all \( x, y, t \),
\[ P(t x, t y) = t^n P(x, y), \]

b) For all \( u_1, u_2, \ldots, u_k \),
\[ P\left( u_1 + \ldots + u_k, u_1 \right) + P\left(u_1 + \ldots + u_k, u_2 \right) + \ldots + P\left( u_1 + \ldots + u_k, u_k \right) = 0. \]

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