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$f(x)$ is a $5^\text{th}$-degree polynomial such that $f(1)=2,$ $f(2)=3,$ $f(3)=4,$ $f(4)=5,$ $f(5)=6,$ and $f(8)=7.$

If the value of $f(9)$ can be expressed as $\frac{a}{b}$ for coprime positive integers $a$ and $b$, find the value of $a+b$.

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