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Let this common value be $ k $. Then,
\frac{x^2}{y} = \frac{y^2}{z} \Rightarrow x^2 z = y^3, \quad \frac{y^2}{z} = \frac{z^2}{x} \Rightarrow y^2 x = z^3.
From these, we deduce a symmetry: suppose $ x = y = z $. Then $ x + y + z = 3x = 1 \Rightarrow x = \frac{1}{3} $. Substituting into $ S $:

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