
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 with two variables that satisfy the following conditions:
1. \( P \) is homogeneous of degree \( n \), i.e., for all real numbers \( t, x, y \), \( P(t x, t y) = t^{n} \cdot P(x, y) \);
2. For all real numbers \( a, b, c \), \( P(a+b, c) + P(b+c, a) + P(c+a, b) = 0 \);
3. \( P(1,0)=1 \).

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