A problem by Brian Yao

Level pending

Let $$N$$ be the sum of the real roots of the polynomial function

$$P(x)=x^{6}+2x^{5}+x^{4}-7x^{3}-18x^{2}-9x-18$$

$$\frac{1}{N}$$ can be expressed in the form $$a^{\frac{4}{3}}+b^{\frac{1}{3}}+c$$, where $$a$$, $$b$$, and $$c$$ are positive integers. Find $$a+b+c$$. Do not make any unwarranted assumptions about the values of $$a$$, $$b$$, and $$c$$.

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