A system has an installed cost of $$150,000$ and an additional cost of $$50,000$ after $10$ years. The annual software maintenance contract cost is $$5000$ for the first $4$ years and $$8000$ there after. In addition, there is expected to be a recurring major upgrade cost of $$15,000$ every $13$ years. Assume that i is $5\%$ per year for county funds.