Trio of abc

Both roots of $x^{2}-(a^{2}+b^{2}+c^{2})x+(a^{2}b^{2}+b^{2}c^{2}+c^{2}a^{2})=0$ are non real.

If $a^{2}+b^{2}+c^{2}<[k](|ab|+|bc|+|ca|)$ ,then find the minimum value of $k$ .

Here, $[•]$ represents greatest integer function and $|•|$ reprents modulus function.