21987_WormonanApple

An apple has the shape of a perfect sphere with the radius equal to 1. There is a worm located at some point A on the surface of the apple. The worm wants to move to the point B, also located on the surface of the apple. The worm may move in two fashions:

- to crawl along the surface of the apple with the constant speed v₀;

- to gnaw a wormhole in the shape of a straight line segment from point A to point B.

The inside of the apple consists of a core surrounded by the pulp. The core of the apple is also a perfect sphere, with the radius equal to P, and the center in the center of the whole apple. The worm can gnaw through the core with the speed v₁ and through the pulp with the speed v₂.

Determine the minimal time the worm needs to get from the point A to the point B. The size of the worm can be neglected.

The first line contains the values of P (0 < P < 1, the value is given with up to 3 decimal places), v₀, v₁, and v₂ (integers in the range from 1 to 100). The second and the third line give the points A and B, respectively, as two angles in the spherical coordinate system. The first angle, the azimuth φ (0 ≤ φ < 360°), gives the angle between the X axis and the projection on the XY plane of the segment connecting the origin to the corresponding point. The second angle, the polar θ (0 ≤ θ ≤ 180°), gives the angle between the Z axis and the segment connecting the origin to the corresponding point. The values φ and θ are integers and given in degrees. In all test cases where θ = 0 or θ = 180, φ is given as 0.

The first and only line of the output should contain the minimal time of travel, accurate to 10^-9.

Sample #1
0.5 10 15 20
0 0
0 180

Sample #2
0.5 20 5 2
0 0
0 180

Sample #1
0.116666667

Sample #2
0.157079633