
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
If \( a, b, c, \) and \( d \) are the roots of the equation \( x^4 - 2x - 6 = 0 \), find the equation whose roots are defined as \( \frac{2(a + b + c)}{d^3}, \frac{2(a + b + d)}{c^3}, \frac{2(a + c + d)}{b^3}, \frac{2(b + c + d)}{a^3} \).

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