Processing the life of a request

Have you ever played a wooden labyrinth board game, a maze puzzle with a steel ball bearing? You might have played it in your childhood. It was a very crazy game. The goal of this game is to send all the steel ball bearings to the center of the wooden labyrinth board through interlinked curvy paths, and these curvy paths have cuts leading to a second curve near the center. All the balls need to navigate to the center of the wooden labyrinth board through these cuts between the curvy paths. If one steel ball reaches the center, then we have to be careful about this ball so that it does not move away from the center when trying to move another ball to the center. You can see this in the following diagram:

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