Integer polynomial

Algebra Level 5

Consider all polynomials with integral coefficients

$a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}=0,$

such that $\frac{4}{3}$ is one of its roots, $3 | a_0,$ and $4 | a_n$ .

Over all such polynomials, find the smallest positive value of $a_n + a_0$ .