After a moment of silence

$x, y, z \in [1; 3]$ are three numbers such that $x + y + z = 6$ . The minimum value of $f(x, y, z) = \dfrac{x^3 + y^2 + z}{3x + 2y + z}$ is

$\large \frac{1}{3}\left(\sqrt[3]{\frac{b + c\sqrt{d}i}{e_1}} + f_1 + \frac{e_2}{\sqrt[3]{\frac{b + c\sqrt{d}i}{e_1}}}\right)^2 + \frac{2}{3}\left(\sqrt[3]{\frac{b + c\sqrt{d}i}{e_1}} + f_1 + \frac{e_2}{\sqrt[3]{\frac{b + c\sqrt{d}i}{e_1}}}\right) + f_2$

Calculate the value of $e_1e_2(b + cd^2) + f_1 + f_2$ .

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Notation:

• $i$ denotes the imaginary unit .
• $d$ is square-free and $\gcd(b, e_1) = \gcd(c, e_1) = \gcd(b, e_2) = \gcd(c, e_2) = 1$ .