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Algebra Level 5

Let f(x)f(x) denote a 7th7^\text{th} degree polynomial satisfying fn=3nf_{n}=3^{n} for n=0,1,2,3,4,5,6,7n=0,1,2,3,4,5,6,7. Find f(8)f(8).


Inspirations: First link, Second link, Third link.

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