A weighted quadratic reciprocal bound with symmetric sums

prove that ${\it \sum} \left( {\frac {1}{101\,{x}^{2}+yz}} \right) {\it \sum} \left( {x}^{2} \right) \geq \frac{1}{34}\,{\frac { \left( {\it \sum} \left( x \right) \right) ^{2}}{{\it \sum} \left( yz \right) }}$ given $x,y,z>0$

Source: InternLM Lean-Workbook, record lean_workbook_5653 (Apache-2.0). Complete source proposition preserved; proof developed independently.