Question

Question: If z1 , z2 ,z3 are complex numbers and |z1|=|z2|=|z3|=1 and (z1)^3+(z2)^3 + (z3)^3 +z1*z2*z3=0 find the maximum and minimum values of |z1+z2+z3|.

My thoughts on the question:
Let z1=a , z2=b and z3=
We know that |a+b+c| <= |a| + |b+c|

|a|=1

So |a+b+c|<= 1 + |b+c|

|b+c|<=|b|+ |c|
|b+c|<= 2

So |a+b+c|<= 3
|a+b+c|max=3


Above is how I first solved this.But as I had another look at the given equation I realized that taking the magnitudes z1,z2,z3 all as 1 ,violates the given equation.And also realized that the equation is true only when magnitudes two of the three complex numbers are 1.

So what should be the maximum and the minimum?

I'm looking for a direct method except from plugging in values and getting the result.