RO-NMO 2000 Problem 102

Demonstrate that if $z_1,z_2\in\mathbb{C}^*$ satisfy the relation: $$z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} ,$$ then $z_1^6=z_2^6$