
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

Determine the polynomial $P$ that satisfies the following conditions:
(a) $P$ is a homogeneous polynomial of degree $n$ in $x$ and $y$, that is, for all real numbers $t, x, y$, $P(t x, t y)=t^{n} P(x, y)$, where $n$ is a positive integer;
(b) For all real numbers $a, b, c$, $P(a+b, c)+P(b+c, a)+P(c+a, b)=0$;
(c) $P(1,0)=1$.

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