For a complex number z, let Re(z) denote that real part of z. Let S be the set of all complex numbers z satisfying $${z^4} - |z{|^4} = 4i{z^2}$$, where i = $$\sqrt { - 1} $$. Then the minimum possible value of |z₁ $$-$$ z₂|², where z₁, z₂$$ \in $$S with Re(z₁) > 0 and Re(z₂) < 0 is .........

Let $$\omega \ne 1$$ be a cube root of unity. Then the minimum of the set $$\{ {\left| {a + b\omega + c{\omega ^2}} \right|^2}:a,b,c$$ distinct non-zero integers} equals ..................

For any integer k, let $${a_k} = \cos \left( {{{k\pi } \over 7}} \right) + i\,\,\sin \left( {{{k\pi } \over 7}} \right)$$, where $$i = \sqrt { - 1} \,$$. The value of the expression $${{\sum\limits_{k = 1}^{12} {\left| {{\alpha _{k + 1}} - {a_k}} \right|} } \over {\sum\limits_{k = 1}^3 {\left| {{\alpha _{4k - 1}} - {\alpha _{4k - 2}}} \right|} }}$$ is

If z is any complex number satisfying $$\,\left| {z - 3 - 2i} \right| \le 2$$, then the minimum value of $$\left| {2z - 6 + 5i} \right|$$ is