
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
The quadratic polynomials \( P(x)=x^{2}+\frac{x}{2}+b \) and \( Q(x)=x^{2}+c x+d \) with real coefficients are such that \( P(x) Q(x) = Q(P(x)) \) for all \( x \). Find all real roots of the equation \( P(Q(x)) = 0 \).

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