# A rational game ends

Algebra Level 5

A polynomial with integer coefficients $$P(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}$$, with $$a_{m}$$ and $$a_{0}$$ being positive integers, has one of the roots $$\dfrac{2}{3}$$. Find the $$n^\text{th}$$ smallest $$(n \geq 10)$$possible value of $$a_{0}+a_{m}$$.

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