
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
Given the quadratic equations:
$$
x^{2}+p x+q=0 \quad \text{and} \quad x^{2}+p^{\prime} x+q^{\prime}=0
$$

with roots \(\alpha, \beta\) and \(\alpha^{\prime}, \beta^{\prime}\), respectively, express the product

$$
\left(\alpha-\alpha^{\prime}\right)\left(\alpha-\beta^{\prime}\right)\left(\beta-\alpha^{\prime}\right)\left(\beta-\beta^{\prime}\right)
$$

as a function of the coefficients \(p, q, p^{\prime}, q^{\prime}\).

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