
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
It is known that the equation \( x^{3} - x - 1 = 0 \) has a single real root \( x_{0} \). Come up with an equation of the form

\[ a \cdot z^{3} + b \cdot z^{2} + c \cdot z + d = 0 \]

where \( a, b, c, d \) are integers and \( a \neq 0 \), for which one of the roots is

\[ z = x_{0}^{2} + 3 \cdot x_{0} + 1 \]

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