Let \((a,b,c) = v, (a',b',c') = v'\)and \(a,a',b,b',c,c'\) be non-negative integers such that \(f(x,y,z) = x^{2} + y^{2} - z^{2} = 0\) whenever \(f\) is evaluated at \(v\) or \(v'\) \(v\) is said to lie in the same family as \(v'\) if \(v = nv'\) for \(n\) a positive integer or one's reciprocal. The question i ask is how many distinct families of solutions of \(f(x,y,z) = 0\) are there.

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