# Ray Inside a Polygon

Suppose we have a convex polygon of n vertices. One of the edges have a light emitting point which emits a ray of light in a known direction. The edges of the polygon are made of mirrors. As a result when a ray falls upon an edge, it is reflected to another directon according to the laws of reflection. So after a certain number of reflections we can evaluate the point hitted by the ray. Our challenge here is to evaluate that point. By the way, if the ray falls upon a vertex of the polygon, it is lost forever. The figure may not show the angles properly. But in actual cases, I mean in your program, you should follow the rules of reflection. Input The input session begins with two numbers 3 ≤ v ≤ 10, 0 ≤ n ≤ 1000, denoting the number of vertices in the polygon and the number of reflections to simulate. After that a line will follow containing (x, y), the position of the light emitting point, which will always lie upon an edge (and will never coincide with any vertex) and 0 ≤ t < 360, the angle in degree between x-axis and the ray, measured in anti-clockwise direction, which is always valid. Next there will be v lines each of which will contain a vertex of the polygon ordered to the anti-clockwise direction. All points will be represented in x, y coordinate system with two digits after the decimal point. A pair of zeros declares the end of the input session. Output For every dataset, output the final point the ray hits after n reflections with two digits after the decimal point. Remember, the point should lie upon an edge of the polygon. If the ray is lost forever within n reflections, output ‘lost forever...’. Output for every dataset should start in a new line. Be extra cautious about precision errors. Because if there is a vertex 4.00 and our ray passes through 4.001, we will count it as ‘lost forever...’, as we are concerned with only two digits after the decimal point.

2/2 Sample Input 44 2.00 0.00 0.00 0.00 0.00 4.00 0.00 4.00 4.00 0.00 4.00 40 2.00 0.00 45.00 0.00 0.00 4.00 0.00 4.00 4.00 0.00 4.00 00 Sample Output lost forever... 4.00 2.00