
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
Find the quadratic polynomial $p(x) = x^2 - (p+q)x + pq$ with real coefficients, such that the equation $p(p(x))=0$ has exactly four distinct real roots, and among such polynomials, determine the polynomial for which the product of the roots is maximized. Calculate the value of $p(0)$.
A) $-\frac{1}{16}$
B) $0$
C) $\frac{1}{16}$
D) $\frac{1}{4}$

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