
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
Suppose that \(P(x, y, z)\) is a homogeneous degree 4 polynomial in three variables such that \(P(a, b, c) = P(b, c, a)\) and \(P(a, a, b) = 0\) for all real \(a, b\), and \(c\). If \(P(1, 2, 3) = 1\), compute \(P(2, 4, 8)\). Note: \(P(x, y, z)\) is a homogeneous degree 4 polynomial if it satisfies \(P(k a, k b, k c) = k^{4}P(a, b, c)\) for all real \(k, a, b, c\).

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