A product of quadratic sums bounds a reciprocal ratio sum

Let $a,b,c>0$ . Prove that: $\left( {{a^2} + {b^2} + {c^2}} \right)\left( {\frac{1}{{{a^2}}} + \frac{1}{{{b^2}}} + \frac{1}{{{c^2}}}} \right) \ge \frac{3}{2}\left( {\frac{{b + c}}{a} + \frac{{c + a}}{b} + \frac{{a + b}}{c}} \right)$

Source: InternLM Lean-Workbook, record lean_workbook_12029 (Apache-2.0). Complete source proposition preserved; proof developed independently.