Complex Egyptian Fractions

543i=1v+1w+1z\dfrac{5}{4-3i} = \dfrac{1}{v} + \dfrac{1}{w} + \dfrac{1}{z}

Let v,w,zv,w,z be the Gaussian integers, satisfying the equations above, where 1<v<w<z1< |v| < |w| < |z| and v+w+zv+w+z is real.

What is the least possible value of v2+w2+z2|v|^2 + |w|^2 + |z|^2?

Note: i2=1i^2 = -1, and z|z| is the absolute value of zz.

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