Alternative N has an initial construction cost of $2400000$ and an annual maintenance cost of $15500$ per year. Alternative SA has an initial construction cost of $52000$ and an annual maintenance cost of $125000.$ Alternative SB has an initial construction cost of $52000$ and an annual maintenance cost of $125000.$ The user costs are $625000$ per year for Alternative n$,410000$ for Alternative SA$,$ and $310000$ per year for Alternative SB$.$ The residual value of each alternative after $20$ years is estimated to be $20\%$ of the initial construction cost. Determine which alternative to choose using an interest rate of $7\%$ and a planning horizon of $20$ years.