\(P=abc\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)+2abc=abc\left(\frac{y}{x}+\frac{y}{z}+\frac{x}{y}+\frac{z}{z}+\frac{z}{x}+\frac{z}{y}\right)+2abc\)
\(=abc\left(\frac{y^2\left(x+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)}{xyz}\right)+2abc\)
\(=\frac{y^2\left(x+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)+2xyz}{\left(x+y\right)\left(y+z\right)\left(x+z\right)}=\frac{y^2\left(x+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)+2xyz}{\left(x+z\right)\left(xy+xz+yz+y^2\right)}\)
\(=\frac{y^2\left(x+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)+2xyz}{x^2y+x^2z+xyz+xy^2+xyz+xz^2+yz^2+y^2z}=\frac{y^2\left(x+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)+2xyz}{x^2\left(y+z\right)+z^2\left(x+y\right)+y^2\left(x+z\right)+2xyz}=1\)
