
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

For a cubic polynomial \( P(x) \) with complex roots \( z_{1}, z_{2}, z_{3} \), let

\[ M(P) = \frac{\max \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)}{\min \left(|z_1 - z_2|, |z_1 - z_3|, |z_2 - z_3|\right)} . \]

Over all polynomials \( P(x) = x^3 + ax^2 + bx + c \), where \( a, b, c \) are nonnegative integers at most 100 and \( P(x) \) has no repeated roots, the twentieth largest possible value of \( M(P) \) is \( m \). Estimate \( A = \lfloor m \rfloor \). An estimate of \( E \) earns 

\[
\max \left(0,\left\lfloor 20 - 20 \left| 3 \ln \left(\frac{A}{E}\right) \right|^{1/2} \right\rfloor \right)
\]

points.

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