
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 $f(x) = x^3 - x^2 + 4x + 2$ have three distinct roots. Define another cubic polynomial $g(x) = x^3 + bx^2 + cx + d$ with leading coefficient $1$, where the roots of $g(x)$ are the cubes of the roots of $f(x)$. Determine the ordered triple $(b, c, d)$.

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