Confusing \(x, y, z\)

Algebra Level 5

\[x^2y+y^2z+z^2x=xy^2+yz^2+zx^2=kxyz\]

Find the range of \(k\) for which there is at least one triplet \((x,y,z)\) of non-zero real numbers such that the equation above holds true.

If it is in the form \((-\infty, a]\cup [b,\infty)\), find \(a+b\).

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