Numerical Methods: Euler to Runge–Kutta · Superchallenge problem

For $y' = y$, $y(0) = 1$, Euler's method on $[0, 1]$ gives $A(h) = (1 + h)^{1/h}$, whose error has an expansion $A(h) = e + c_1h + c_2h^2 + O(h^3)$. Using the three values $A(1)$, $A\!\left(\dfrac{1}{2}\right)$ and $A\!\left(\dfrac{1}{4}\right)$, form the combination $\alpha A(1) + \beta A\!\left(\dfrac{1}{2}\right) + \gamma A\!\left(\dfrac{1}{4}\right)$ that reproduces $e$ exactly and cancels both the $h$ and $h^2$ error terms, and evaluate it. Express your answer as a common fraction.