Amy is choosing between two apartments. Apartment A is selling at $9$ million, and apartment $B$ is selling at $Y11$ million. She can take a $30-$year mortgage to finance $70\%$ of the price with an interest rate of $5\mathrm{.}145\%($APR$).$ Other than the down payment, she needs to pay a total transaction fee $($taxes$,$ commission, etc.$)$ equaling $2\%$ of the purchase price. She plans to sell the apartment after $3$ years. In the meantime, she does not plan to live in this apartment, and she can rent it out. Apartment $A$ is expected to generate a rent of $8000$ per month, and apartment B is expected to generate a rent of $12,000$ per month. After $3$ years, she expects to sell apartment A for $11\mathrm{.}5$ million or apartment B for $13$ million, and pre$-$pay the outstanding balance of her mortgage. Suppose her opportunity cost of capital is $7\mathrm{.}2\%($that is$,$ she can invest the cash elsewhere and make $7\mathrm{.}2\%$ per year in term of APR, monthly compounding$).$

$(1)$ What are her monthly payments for the two apartments?

$(2)$ What is the NPV if she purchases apartment A$?($Hint: for each month, mortgage payment is the cash outflow, and the rent is the cash inflow$)$

$(3)$ What is the NPV if she purchases apartment B$?(($Hint: for each month, mortgage payment is the cash outflow, and the rent is the cash inflow$))$

$(4)$ Which apartment should she choose?