\[ \large{f(x,y,z)= 2x^{2}+2y^{2}-2z^{2}+\frac{7}{xy}+\frac{1}{z}} \]

There are three pairwise distinct numbers \(a,b,c\) that satisfies the above equation in a way such that \( f(a,b,c)= f(b,c,a)=f(c,a,b)\).

What is \(a+b+c\)?

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