
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 \( p(x) \) with real coefficients and leading coefficient \( 1 \). The polynomial is called **peculiar** if the equation \( p(p(x)) + x - 1 = 0 \) is satisfied by exactly 4 distinct real numbers. Among all such peculiar quadratic polynomials, there exists a unique polynomial \( \tilde{p}(x) \) for which the product of the roots is minimized. Determine \( \tilde{p}(2) \).
A) 0
B) 1
C) 2
D) 3

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