The answer is not 0

Algebra Level 5

Let $a,b,c$ be non-negative reals. Find the maximum value of $k$ such that $(a+b+c)\left(\frac { 1 }{ a } +\frac { 1 }{ b } +\frac { 1 }{ c } \right)+\frac { k(ab+bc+ca) }{ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } } \ge 9+k.$ Write your answer to 3 decimal places.