
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 two-variable polynomials $P$ that satisfy the following conditions:
1. $P$ is homogeneous of degree $n$. That is, for all real numbers $t$, $x$, and $y$, $P(tx, ty) = t^n P(x, y)$.
2. For all real numbers $a$, $b$, and $c$:
   \[
   P(a+b, c) + P(b+c, a) + P(c+a, b) = 0.
   \]
3. $P(1, 0) = 1$.

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