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 \( a_n + a_0 \).