Integer polynomial

Algebra Level 5

Consider all polynomials with integral coefficients

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

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 $$a_n + a_0$$.

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