Chain Rule

Calculus Level 5

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

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

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 $a+b?$