A symmetric rational correction bounds pairwise products

Let $x,y,z>0$ ,then prove that $6\,zy + 6\,xz + 6\,yx\leq 8\,{\frac {\left( x + y + z \right) xyz}{\left( y + z \right) \left( z + 2\,x + y \right) }} + 8\,{\frac {\left( x + y + z \right) xyz}{\left( z + x \right) \left( x + 2\,y + z \right) }} + 8\,{ \frac {\left( x + y + z \right) xyz}{\left( x + y \right) \left( y + 2\,z + x \right) }} + 3\,{x}^{2} + 3\,{y}^{2} + 3\,{z}^{2}.$

Source: InternLM Lean-Workbook, record lean_workbook_3067 (Apache-2.0). Complete source proposition preserved; proof developed independently.