A rational game ends

Algebra Level 4

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 $\frac{2}{3}$. Find the $n^\text{th}$ smallest $(n \geq 10)$ possible value of $a_{0}+a_{m}$.