Disguised Inequality

Algebra Level 4

\[(a+b+c)(a^2+b^2+c^2)+3abc\geq N(a+b)(a+c)(b+c)\]

If \(a,b,c\) and \(N\) are positive reals such that the maximum value of \(N\) which satisfies the inequality above can be expressed in the form \(\frac{m}{n}\), where \(m\) and \(n\) are coprime positive integers, find the value of \(m+n\).

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