Austin needs to purchase a new heating$/$cooling system for his home. He is thinking about having a geothermal system installed, but he wants to know how long it will take to recoup the additional cost of the system. The geothermal system will cost $$18,500.$ A conventional system will cost $$5,500.$ Austin is eligible for a $30\%$ tax credit to be applied immediately to the purchase only if he chooses the geothermal system. He estimates that he will save $$1,300$ per year in utility bills with the geothermal system. These cash outflows can be assumed to occur at the end of the year. The cost of capital $($or interest rate$)$ for Austin is $8\%.$ How long will Austin have to use the system to justify the additional expense over the conventional model? In other words, what is the DISCOUNTED payback period in years?