A product of weighted cyclic ratio sums is at most 343 over eight

prove that ${\frac {343}{8}}\geq \left( 6\,{\frac {x}{x+y}}+{\frac {y}{y+z}} \right) \left( 6\,{\frac {y}{y+z}}+{\frac {z}{z+x}} \right) \left( 6\,{\frac {z}{z+x}}+{\frac {x}{x+y}} \right)$ where $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_45205 (Apache-2.0). Complete source proposition preserved; proof developed independently.