Dig or Climb

Benjamin Forest VIII is the king of a country. His close friend Nod lives in a village far from the castle. Nod falls seriously ill and his life is in danger, so the king orders his subordinate Red to deliver good medicine as fast as possible. There is no road from the castle to the village, so Red has to climb over mountains and cross canyons to reach it.

Red travels along the shortest path on the map, the straight line between the castle and the village. The cross section of the terrain along that line is a polyline through the $n$ points $(x_1, y_1), \dots, (x_n, y_n)$.

Figure 1. An example route from the castle to the village

$x_i$ is the horizontal distance from the castle to point $i$, and $y_i$ is the height of point $i$. The castle is at $(x_1, y_1)$ and the village is at $(x_n, y_n)$.

Red walks on the surface at speed $v_w$. He also has a skill that cuts a horizontal tunnel through a mountain, and he moves inside a mountain at speed $v_c$.

Red has two ways to move. The first is walking along the polyline, either to the left or to the right. That takes the length of the walked part of the polyline divided by $v_w$. The second is digging through a mountain. If two points $(a, y)$ and $(b, y)$ on the polyline have the same height, $a < b$, and the terrain height is greater than $y$ at every $x$ with $a < x < b$, then Red can move between the two points along the horizontal line. That takes $(b - a) / v_c$. If the terrain height is at most $y$ anywhere strictly between the two points, that stretch is not inside a mountain and cannot be dug.

Write a program that computes the minimum time to reach the village.

The input is a sequence of datasets. Each dataset has the following format.

n
vw vc
x1 y1
...
xn yn

The first line holds the number of points $n$. The second line holds the walking speed $v_w$ and the digging speed $v_c$. Each of the next $n$ lines holds the coordinates $x_i$ and $y_i$ of one point.

$2 \le n \le 1000$, $1 \le v_w, v_c \le 10$, $-10000 \le x_i, y_i \le 10000$, and $x_i < x_j$ whenever $i < j$. Every value in the input is an integer.

The input ends with a line that holds $0$ in place of $n$. That line is not a dataset and must not be processed.

For each dataset, print the minimum time to reach the village on its own line. Round the value to six digits after the decimal point and print all six digits. Print no other character and no extra space.