
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
The equation  $x^3 + ax^2 + bx + c = 0$  has three (not necessarily distinct) real roots  $t, u, v$ . For which  $a, b, c$  do the numbers  $t^3, u^3, v^3$  satisfy the equation  $x^3 + a^3x^2 + b^3x + c^3 = 0$ ?

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