A coprime rational game 3'

Algebra Level 5

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