In a warehouse, there are three types of boxes: small, medium, and large. A small box costs $10 each, a medium box costs $20 each, and a large box costs $40 each. All boxes will get a 15% discount. If a customer buys 5 small boxes, 7 medium boxes, and 3 large boxes, what is the total cost after applying the discount? Additionally, what is the weighted average cost per box? Let's think step by step and output the final answer within \boxed{}.
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