Artisan Bread Boutique bakes four unique varieties of artisanal bread. Each variety undergoes these four stages: (1) dough preparation, (2) baking, (3) quality control, and (4) packaging for distribution. The time required for each stage is outlined in the provided table for each bread type. After a bustling week, the manager calculates that the dough prep team worked for 262 hours, the bakers for 160 hours, the quality control inspectors for 29 hours, and the packaging team for 68 hours. How many loaves of each bread type were baked? (Formulate a system of equations and solve it by utilizing the inverse matrix method.)