
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
Consider a quadratic polynomial with real coefficients and a leading coefficient of $1$, defined as \( p(x) = x^2 - (r+s)x + rs \) where $r$ and $s$ are the roots. A polynomial is termed as *mischievous* if the polynomial $p(p(x)) = 0$ is satisfied by exactly four real numbers. Among all the mischievous quadratic polynomials, find the unique polynomial $\hat{p}(x)$ for which the product of the roots is minimized.

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