Maximize complex numbers

Algebra Level 4

Determine the largest real number k\color{#3D99F6}{k} such that z1z2+z2z3+z3z1kz1+z2+z3\large | z_1z_2 + z_2z_3 + z_3z_1 | \geq \color{#3D99F6}{k} | z_1 + z_2 + z_3| for all complex numbers z1,z2,z3z_1 , z_2 , z_3 with unit absolute value.

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