Alternate Wye Circuit with Same Current Draw

Three AC voltage sources excite three impedances as shown. In the "reference case", $Z_A = Z_{A0}$ , $Z_B = Z_{B0}$ , and $Z_C = Z_{C0}$ . Find the values of $Z_A$ , $Z_B$ , and $Z_C$ which minimize the magnitude of neutral voltage $V_N$ , subject to the following constraints:

1) $Z_A$ must be purely resistive (zero imaginary part)

2) The currents flowing through $Z_A$ , $Z_B$ , and $Z_C$ must be the same (in magnitude and phase angle) as those in the reference case

3) The real (resistive) parts of $Z_A$ , $Z_B$ , and $Z_C$ must not be negative

For this optimal configuration, what is the sum of the real (resistive) parts of $Z_A$ , $Z_B$ , and $Z_C$ ?

Details and Assumptions:

1) $V_A = 100 \, e^{j 0}$
2) $V_B = 100 \, e^{-j 2 \pi/3}$
3) $V_C = 100 \, e^{j 2 \pi/3}$
4) $Z_{A0} = 1 + j 0$
5) $Z_{B0} = 2 + j 1$
6) $Z_{C0} = 3 - j 2$
7) $j = \sqrt{-1}$