Maximize complex numbers

Algebra Level 4

Determine the largest real number \(\color{blue}{k}\) such that \[\large | z_1z_2 + z_2z_3 + z_3z_1 | \geq \color{blue}{k} | z_1 + z_2 + z_3|\] for all complex numbers \(z_1 , z_2 , z_3\) with unit absolute value.

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