
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 parameters \( p \) and \( q \) such that the roots (not necessarily distinct) of equation (1) are real and can be indexed in a way that satisfies (2).

$$
x^{4} + p x^{2} + q = 0
$$

$$
\frac{x_{2}}{x_{1}} = \frac{x_{3}}{x_{2}} = \frac{x_{4}}{x_{3}}
$$

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