Determine the smallest number $M$ such that the inequality
$$\begin{vmatrix}ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\end{vmatrix}\le M\bigl(a^2+b^2+c^2\bigr)^2$$
holds for all real numbers $a,b,c.$ The original answer is in the format \frac{m}{n}\sqrt{2}, please give the value of m + n.

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