
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
Determine the value of the parameter \( p \) such that the equation

$$
x^2 + px + p - 1 = 0
$$

satisfies the requirement for its roots \( x_1 \) and \( x_2 \) given by

$$
x_1^2 + x_1^3 = -\left(x_2^2 + x_2^3\right).
$$

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