A rational game ends

Algebra Level 4

A polynomial with integer coefficients P(x)=amxm+am1xm1++a0P(x)=a_{m}x^{m}+a_{m-1}x^{m-1}+\cdots+a_{0}, with ama_{m} and a0a_{0} being positive integers, has one of the roots 23\frac{2}{3}. Find the nthn^\text{th} smallest (n10)(n \geq 10) possible value of a0+ama_{0}+a_{m}.


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