Inspired by Daniel Liu

Algebra Level 5

Consider all sets of real numbers $$x, y, z$$ which satisfy the equation

$xy^2 + yz^2 + zx^2 = 3$

Let the infimum value of $$\frac{ x ^2 + y^2 } { z^2 } + \frac{ y^2 + z^2 } { x^2 } + \frac{ z^2 + x^2 } { y^2 }$$ be denoted as $$n$$.

For how many sets of ordered triples $$( x, y, z )$$ which satisfy the initial equation, also satisfy

$\frac{ x ^2 + y^2 } { z^2 } + \frac{ y^2 + z^2 } { x^2 } + \frac{ z^2 + x^2 } { y^2 } = n ?$

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