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 \(a_{0}+a_{m}\).