Inequality

Algebra

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 $\alpha+\beta$ .