
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
Let \( f(x), g(x) \), and \( h(x) \) be polynomials with real coefficients, and let \( f^{2}(x) = x g^{2}(x) + x h^{2}(x) \). Then \( f(x) = g(x) = h(x) = 0 \). What is the conclusion when \( f(x), g(x), h(x) \) are polynomials with complex coefficients?

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