Three of the four endpoints of the axes of an ellipse are, in some order, \[(-2, 4), \; (3, -2), \; (8, 4).\]Find the distance between the foci of the ellipse.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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