CF1807F Bouncy Ball

You are given a room that can be represented by a $ n \times m $ grid. There is a ball at position $ (i_1, j_1) $ (the intersection of row $ i_1 $ and column $ j_1 $ ), and it starts going diagonally in one of the four directions: - The ball is going down and right, denoted by $ \texttt{DR} $ ; it means that after a step, the ball's location goes from $ (i, j) $ to $ (i+1, j+1) $ . - The ball is going down and left, denoted by $ \texttt{DL} $ ; it means that after a step, the ball's location goes from $ (i, j) $ to $ (i+1, j-1) $ . - The ball is going up and right, denoted by $ \texttt{UR} $ ; it means that after a step, the ball's location goes from $ (i, j) $ to $ (i-1, j+1) $ . - The ball is going up and left, denoted by $ \texttt{UL} $ ; it means that after a step, the ball's location goes from $ (i, j) $ to $ (i-1, j-1) $ . After each step, the ball maintains its direction unless it hits a wall (that is, the direction takes it out of the room's bounds in the next step). In this case, the ball's direction gets flipped along the axis of the wall; if the ball hits a corner, both directions get flipped. Any instance of this is called a bounce. The ball never stops moving. In the above example, the ball starts at $ (1, 7) $ and goes $ \texttt{DL} $ until it reaches the bottom wall, then it bounces and continues in the direction $ \texttt{UL} $ . After reaching the left wall, the ball bounces and continues to go in the direction $ \texttt{UR} $ . When the ball reaches the upper wall, it bounces and continues in the direction $ \texttt{DR} $ . After reaching the bottom-right corner, it bounces once and continues in direction $ \texttt{UL} $ , and so on. Your task is to find how many bounces the ball will go through until it reaches cell $ (i_2, j_2) $ in the room, or report that it never reaches cell $ (i_2, j_2) $ by printing $ -1 $ . Note that the ball first goes in a cell and only after that bounces if it needs to.

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