A Flat Tangent on an Interval

Let $f(x) = \mathrm{e}^x(a - x) + x + 1$.

1. If the graph of $f(x)$ has a tangent line of slope zero at some point with $x$-coordinate in $[0, 1]$, find the range of possible values of $a$. 2. When $a = 1$, determine the number of zeros of $f(x)$, justifying your answer.