
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 quadratic polynomials \(P(x)\), \(Q(x)\), and \(R(x)\) such that each of the equations \(P(x) + Q(x) = 0\), \(P(x) + R(x) = 0\), \(Q(x) + R(x) = 0\), and \(P(x) + Q(x) + R(x) = 0\) has exactly two roots. The products of the roots of the first three equations are \(r\), \(q\), and \(p\) respectively. What is the product of the roots of the fourth equation?

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