
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 binary polynomials \( P \) that satisfy the following conditions:
(1) \( P \) is a homogeneous polynomial of degree \( n \);
(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}
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