Twin Tangents

Let $f(x) = a\mathrm{e}^x$ and $g(x) = \ln x + b$ with $a, b \in \mathbb{R}$.

1. When $b = 1$, if $f(x) \geqslant g(x)$ holds for all $x > 0$, find the range of possible values of $a$. 2. Let $l_1, l_2$ be two tangent lines to the curve $y = g(x)$ whose slopes multiply to $1$.

1. Let $x_0$ be the $x$-coordinate of the intersection point of $l_1$ and $l_2$. Prove that $x_0 < 1$.
2. If $l_1$ and $l_2$ are also tangent to the curve $y = f(x)$, find the relation between $a$ and $b$, and the range of possible values of $b$.