The Answer is Not \( \frac{ 3}{ \sqrt{2} } \)

Algebra Level 5

Over all positive triples of real numbers, what is the largest value of \(k\) (to 2 decimal places) such that

\[ \sqrt{ \frac{a} { b+c} } + \sqrt{ \frac{b}{ c + a }} + \sqrt{ \frac{ c}{a+b} } \geq k ?\]

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