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I would reinforce the perimeter of the cutout on the inside of the seat with 3/4" thick mahogany strips . I would use the widest pieces that will fit inside. If only narrow pieces will fit I would substitute mahogany with white oak and go thicker. Glue and screw everything together and it should be plenty strong. Another fix would be to cut the entire perimeter reinforcement out of a single piece of marine plywood and install it on the inside same as the wood strips. This method would probably be the strongest as there would be no joints at the corners. It depends on how much reinforcement you need.