The Octopus Polynomial

Algebra Level 4

Suppose f(x)f(x) is a degree-88 polynomial such that f(2i)=12if(2^i)=\frac{1}{2^i} for all integers 0i8 0 \leq i \leq 8. If f(0)=abf(0)= \frac{a}{b}, where aa and b b are coprime positive integers, what is the value of a+b?a+b?

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