A mobile phone manufacturer is exploring ways to lower costs in its assembly process. The company is considering the purchase and installation of automated systems to handle the phone battery installation, which could reduce labor, material, and energy costs. The cost of procuring and installing the automation equipment is $$500,000,$ with an estimated useful life of $3$ years. A full$-$time technician will be required for ongoing maintenance and programming at a cost of $$65,000$ per year. With the automated system in place, the company can reallocate $4$ full$-$time employees who currently handle battery installation. These employees each earn $$55,000$ per year. Additionally, there is a quality control inspector earning $$52,000$ per year who will still be required after automation. The current cost of battery adhesives is $$0\mathrm{.}6$ per unit, and the energy cost is $$0\mathrm{.}03$ per unit. With the new system, both costs will be reduced by $40\%.$ How many mobile phones should be produced and sold over the next three years to justify the investment in the automation system?