The following equation represents the continuous topography of a mountainous region, giving the elevationheight above sea level $h$ at any point $(x,y)$: $$h=(5000-\frac{x^2+y^2+xy}{200}+\frac{25(x+y)}{2})\cdot e^{-|\frac{x^2+y^2}{1000000}-\frac{3(x+y)}{2000}+\frac{7}{10}|}.$$

A mosquito intends to fly from $A(200,200)$ to $B(1400,1400)$, without leaving the area given by $0\le x,y\le 1600$.

Because of the intervening mountains, it first rises straight up to a point $A^{\prime }$, having elevation $f$. Then, while remaining at the same elevation $f$, it flies around any obstacles until it arrives at a point $B^{\prime }$ directly above $B$.

First, determine $f_{\min }$ which is the minimum constant elevation allowing such a trip from $A$ to $B$, while remaining in the specified area.
Then, find the length of the shortest path between $A^{\prime }$ and $B^{\prime }$, while flying at that constant elevation $f_{\min }$.

Give that length as your answer, rounded to three decimal places.

Note: For convenience, the elevation function shown above is repeated below, in a form suitable for most programming languages:
h=( 5000-0.005*(x*x+y*y+x*y)+12.5*(x+y) ) * exp( -abs(0.000001*(x*x+y*y)-0.0015*(x+y)+0.7) )