
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
Let  $P(x) = x^3 + ax^2 + bx + 1$  be a polynomial with real coefficients and three real roots  $\rho_1$ ,  $\rho_2$ ,  $\rho_3$  such that  $|\rho_1| < |\rho_2| < |\rho_3|$ . Let  $A$  be the point where the graph of  $P(x)$  intersects  $yy'$  and the point  $B(\rho_1, 0)$ ,  $C(\rho_2, 0)$ ,  $D(\rho_3, 0)$ . If the circumcircle of  $\vartriangle ABD$  intersects  $yy'$  for a second time at  $E$ , find the minimum value of the length of the segment  $EC$  and the polynomials for which this is attained.

*Brazitikos Silouanos, Greece*

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