An Isosceles on an Exponential

Let $f(x) = e^x - ax + a$ (with $a \in \mathbb{R}$), and suppose its graph meets the $x$-axis at $A(x_1, 0)$ and $B(x_2, 0)$ with $x_1 < x_2$.

1. Find the range of possible values of $a$. 2. Prove that $f'\left(\sqrt{x_1x_2}\right) < 0$. 3. Let $C$ be a point on the graph of $y = f(x)$ such that $\triangle ABC$ is an isosceles right triangle, and set $t = \sqrt{\dfrac{x_2 - 1}{x_1 - 1}}$. Find the value of $(a - 1)(t - 1)$.