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.The answer is in the form k\sqrt{m}+n,. Please provide the value of k + m + n.

Let's think step by step and output the final answer within \boxed{}.
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