Inequality

Algebra Level 4

If a,b,c0a,b,c\geq 0, the maximum value of NN which satisfies the inequality

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

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

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