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If a,b,c≥0a,b,c\geq 0a,b,c≥0, the maximum value of NNN 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(a+b+c)(ab+bc+ca)(a+b)(b+c)(c+a)≥N
can be expressed in the form αβ\frac{\alpha}{\beta}βα, where α\alphaα and β\betaβ are coprime, positive integers. Find the value of α+β\alpha+\betaα+β.
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