Sophia's Ceramics produces figurines for novelty gift shops in tourist areas. Each week she can produce $1000$ units of a design for $$8,000$ in costs, but many must be discarded due to imperfections. She adjusts the process and eliminates more imperfections each week. She can expect her yield of sellable figurines to be $50\%$ the first week but to increase by $2\%$ each.What is her initial cost per figurine in the $16$th week?$$

Sophia must set a price before she starts producing. If she expects a total demand of $10,000$ units, what is her break$-$even price?