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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