*Harmonic Points*

Here is the definition of Harmonic Points : If point $A$ and $B$ are on the graph of function $f(x)$ and $g(x)$ respectively and $A$ and $B$ are symmetric along the $x$ -axis, then $A$ and $B$ are called a pair of Harmonic Points for the function $f(x)$ and $g(x)$ .

If $f(x)=\begin{cases} 2xe^{x+1}, & x \leq 0 \\ \dfrac{x^2}2-x, & x>0 \end{cases}$ , $g(x)= \dfrac{\sqrt{e}}{2}-ax$ , and $f(x)$ and $g(x)$ have exactly $4$ pairs of Harmonic Points , find the range of $a$ .

The range can be expressed as $(l,r)$ . Submit $\lfloor 1000(2r-l) \rfloor$ .