A cyclic quartic inequality with asymmetric coefficients

Prove that for non-negative real numbers x, y, and z, the following inequality holds: $49\, \left( x+y+z \right) ^{4}-475\, \left( x+y+z \right) ^{2} \left( xy+xz+yz \right) -408\, \left( x+y+z \right) xyz+288\, \left( x+y+z \right) \left( {x}^{2}y+{y}^{2}z+{z}^{2}x \right) +832\, \left( xy+xz+yz \right) ^{2}\geq 0$

Source: InternLM Lean-Workbook, record lean_workbook_5981 (Apache-2.0). Complete source proposition preserved; proof developed independently.