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Consider all polynomials with integral coefficients
anxn+an−1xn−1+⋯+a1x+a0=0, a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}=0,anxn+an−1xn−1+⋯+a1x+a0=0,
such that 43\frac{4}{3}34 is one of its roots, 3∣a0,3 | a_0,3∣a0, and 4∣an4 | a_n4∣an.
Over all such polynomials, find the smallest positive value of an+a0 a_n + a_0 an+a0.
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