
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 \( P, Q, R \) be polynomials and let \( S(x) = P\left(x^{3}\right) + x Q\left(x^{3}\right) + x^{2} R\left(x^{3}\right) \) be a polynomial of degree \( n \) whose roots \( x_{1}, \ldots, x_{n} \) are distinct. Construct with the aid of the polynomials \( P, Q, R \) a polynomial \( T \) of degree \( n \) that has the roots \( x_{1}^{3}, x_{2}^{3}, \ldots, x_{n}^{3} \).

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