
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
Quadratic polynomials \( P(x) \) and \( Q(x) \) with real coefficients are such that together they have 4 distinct real roots, and each of the polynomials \( P(Q(x)) \) and \( Q(P(x)) \) has 4 distinct real roots. What is the minimum number of distinct real numbers that can be among the roots of the polynomials \( P(x) \), \( Q(x) \), \( P(Q(x)) \), and \( Q(P(x)) \)?

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