
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
Find all polynomials \( P(x, y) \) in two variables such that: 
1. \( P(tx, ty) = t^n P(x, y) \) for some positive integer \( n \) and all real \( t, x, y \);
2. for all real \( x, y, z \): \( P(y + z, x) + P(z + x, y) + P(x + y, z) = 0 \);
3. \( P(1, 0) = 1 \).

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