Complicated Inequality

Algebra Level 5

Find the maximum value of NN which satisfies the following inequality over all positive reals a,ba,b and cc:

abc(a+b+c)+(a+b+c)2N3abc(a+b+c).\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})+(a+b+c)^2\geq N\sqrt{3abc(a+b+c)}.

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