Algebra Level 4

Let $$f(x) = ax^2 + bx + c$$, where $$a,b,c$$ are integers. Suppose that $$f(1)=0, 50<f(7)<60, 70<f(8)<80$$ and$$500m<f(100)<500(m+1)$$ for some integer $$m$$.

Also, let the non-integer root of $$f(x)$$ be $$n$$. Find $$m+2n$$.

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