CF724C Ray Tracing

There are $ k $ sensors located in the rectangular room of size $ n×m $ meters. The $ i $ -th sensor is located at point $ (x_{i},y_{i}) $ . All sensors are located at distinct points strictly inside the rectangle. Opposite corners of the room are located at points $ (0,0) $ and $ (n,m) $ . Walls of the room are parallel to coordinate axes. At the moment $ 0 $ , from the point $ (0,0) $ the laser ray is released in the direction of point $ (1,1) $ . The ray travels with a speed of  meters per second. Thus, the ray will reach the point $ (1,1) $ in exactly one second after the start. When the ray meets the wall it's reflected by the rule that the angle of incidence is equal to the angle of reflection. If the ray reaches any of the four corners, it immediately stops. For each sensor you have to determine the first moment of time when the ray will pass through the point where this sensor is located. If the ray will never pass through this point, print $ -1 $ for such sensors.

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In the first sample, the ray will consequently pass through the points $ (0,0) $ , $ (1,1) $ , $ (2,2) $ , $ (3,3) $ . Thus, it will stop at the point $ (3,3) $ after $ 3 $ seconds. In the second sample, the ray will consequently pass through the following points: $ (0,0) $ , $ (1,1) $ , $ (2,2) $ , $ (3,3) $ , $ (2,4) $ , $ (1,3) $ , $ (0,2) $ , $ (1,1) $ , $ (2,0) $ , $ (3,1) $ , $ (2,2) $ , $ (1,3) $ , $ (0,4) $ . The ray will stop at the point $ (0,4) $ after $ 12 $ seconds. It will reflect at the points $ (3,3) $ , $ (2,4) $ , $ (0,2) $ , $ (2,0) $ and $ (3,1) $ .