\(\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)^2\le3\left(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}\right)\Rightarrow\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\le\sqrt{3}\)
\(P=\frac{1}{x+x+y}+\frac{1}{y+y+z}+\frac{1}{x+z+z}\)
\(P\le\frac{1}{9}\left(\frac{1}{x}+\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}+\frac{1}{x}+\frac{1}{z}+\frac{1}{z}\right)=\frac{1}{3}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)=\frac{\sqrt{3}}{3}\)
\(\Rightarrow P_{max}=\frac{\sqrt{3}}{3}\) khi \(x=y=z=\sqrt{3}\)
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