Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
6\sqrt{3}+6
Let $z_1,$ $z_2,$ $z_3,$ and $z_4$ be the four distinct complex solutions of the equation
\[
  z^4 - 6z^2 + 8z + 1 = -4(z^3 - z + 2)i.
\]Find the sum of the six pairwise distances between $z_1,$ $z_2,$ $z_3,$ and $z_4$ in the complex plane.
train