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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