# Calculus *and* Algebra

Calculus Level 5

The function $$F(x)$$ is defined by the following identity:

$$F\left((F(x)+x)^k\right) = (F(x)+x)^2-x$$

The value of $$F(1)$$ is such that a finite number of possible numerical values of $$F'(1)$$ can be determined solely from the above information. The maximum value of $$k$$ such that $$F'(1)$$ is an integer can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are coprime integers. What is the value of $$a+b$$?

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