Inequality

Algebra Level 4

If \(a,b,c\geq 0\), the maximum value of \(N\) which satisfies the inequality

\[\frac{(a+b)(b+c)(c+a)}{(a+b+c)(ab+bc+ca)}\geq N\]

can be expressed in the form \(\frac{\alpha}{\beta}\), where \(\alpha\) and \(\beta\) are coprime, positive integers. Find the value of \(\alpha+\beta\).

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