Reciprocal sum of two roots

Let function $f(x)=e^x-a(x-1)$ , $x \in \mathbb R$ , $a$ is a parameter and $a \in \mathbb R$ .

If $f(x)=0$ has two distinct roots $x_1, x_2$ , is it always true that $\dfrac{1}{x_1}+\dfrac{1}{x_2}>1$ ?

Hint: $f(x)$ does not necessarily has two distinct roots for every possible $a \in \mathbb R$ . What is the range of $a$ so that $f(x)$ has two roots?