World Championship in "Highlander"

At the LCS, the world championship final for the game "Highlander" is being broadcast live! A screen and projector have been set up at the "Coliseum," drawing a massive crowd of fans. Unfortunately, I arrived late, and the game was already underway.

I quickly visualized a coordinate system where the screen is represented by the segment $AB$, with endpoints at $(0,0)$ and $(w,0)$. The image is projected from a point located at (\frac{w}{2}, 5000), and I am positioned at $(x,y)$. From point $C$, one can view the game if and only if point $C$ has a positive y-coordinate, and the angles $\angle ABC$ and $\angle BAC$ do not exceed 135^\circ.

I am free to move around the field, but I cannot cross the screen. What is the minimum distance I need to travel to reach a point where I can watch the game?

Output

Output the minimum distance I need to travel to watch the game, with a precision of at least four decimal places.