\[S = \left \{\left(\frac{x+y-z}{x+y+z}\right)^2 + \left(\frac{x-y+z}{x+y+z}\right)^2 + \left(\frac{-x+y+z}{x+y+z}\right)^2: x,y,z \in \mathbb R^+ \right \}\]

Let \(S\) be a set defined as above, where \(x, y, z\) are positive real numbers.

If \(a = \text{sup } S \) \((\)supremum of \(S)\) and \(b = \text{inf } S \) \((\)infimum of \(S),\) compute \(\frac{a}{b}\).

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