Black Co$.$ is considering a $$10$ million project that last five years, implying straight$-$line depreciation per year of $$2$ million. The cash revenues less cash expenses per year are $$3,500,000.$ The corporate tax rate is $21$ percent. The risk$-$free rate is $10$ percent, and the cost of unleveraged equity is $20$ percent. The present values of a $5-$year annuity of $$1$ per year with discounted rate of $10\%$ and $20\%$ are $3\mathrm{.}7908$ and $2\mathrm{.}9906,$ respectively. The present values of $$1$ to be received after $5$ years with discounted rate of $10\%$ and $20\%$ are $0\mathrm{.}6209$ and $0\mathrm{.}4019,$ respectively. Additional information is as follows:

$(1)$ Black Co$.$ can obtain a $5-$year, nonamortizing loan for $$7,500,000$ after flotation costs at the risk$-$free rate of $10\%.$ The flotation cost is $1$ percent of the gross proceeds of its loan.

$(2)$ Suppose that the project of Black Co$.$ is deemed socially beneficial and the state of New Jersey grants the firm a $$7,500,000$ loan at $8\%$ interest. The flotation costs are absorbed by the state.

Requirements:

$(1)$ What is the value of the project if the project is financed with all equity?

$(2)$ If Black Co$.$ also finances its project with debt, $($a$)$ how much are total debt needed to be financed and its flotation costs $?$

$(3)$ What are the NPV of the loan and the adjusted NPV when considering the debt financing to conduct the project?

$(4)$ Alternatively, if the debt is borrowed from the New Jersey, what are the value of loan from the state and the adjust NPV of the project $?$