Let $f(x) = x + e^x - 1$, and let $f^{-1} (x)$ denote the inverse function of $f(x)$.

Then the integral $\displaystyle \int_{e}^{1 + e^2} f^{-1} (x) \, dx$ evaluates to $k_0 + k_1 e^1 + k_2 e^2 + \cdots + k_n e^n ,$ where $k_0, k_1, \ldots, k_n$ are rational numbers and $e\approx 2.718$ is Euler's number.

If the sum $k_0 + k_1 + k_2 + \cdots + k_n$ can be expressed as $\frac{A}{B},$ where $A$ and $B$ are coprime positive integers, what is $A+B+n?$

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