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@billy 's questions are relevant, please edit your questions to add the answers (even if you can't answer every question). The most important one is :

Can you both turn and move forward at the same time?

You can use the navigation stack to generate trajectories without moving your robot thanks to the service make_plan, that will return a path (i.e. a list of poses). Since you can only use 4 commands you can call the service to get the positions and then try to reach each of them. Since you said you are using a map previously created, you need to use AMCL to get the position of the robot. By comparing the robot pose with the goal you have to reach you could decide in which direction your robot has to move. You can then decide to have a goal tolerance to decide to reach the next goal.