
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  $S(x) = P(x^3) + xQ(x^3) + x^2R(x^3)$  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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