$4.$ increases in gold prices have prompted TomKat Goldfields, Inc. to consider reopening an inactive mine. Last year, TomKat paid a team of geologists $$400,000$ to assess the amount of recoverable gold at the mine. The geologists estimated that $18,000$ troy ounces of gold could still be recovered from the mine. The price of gold is now $$1800$ per ounce, and hedge contracts can be used to lock in this price for up to five years. Direct excavation costs would be $$5\mathrm{.}2$ million per year for five years. These expenditures would allow recovery of $2,000$ ounces of gold the first year, and $4,000$ ounces in each of the second through fifth years. New mining equipment with a cost of $$8$ million would be needed right away. The equipment could be depreciated according to the five$-$year MACRS schedule, and would be sold at the end of year $5$ for an estimated salvage value of $$2\mathrm{.}0$ million. The tax rate is $21\%.$

The geologists$$ report noted that TomKat has sufficient furnace capacity at its platinum plant, as Tomkat$$s platinum division uses only $60\%$ of the furnace capacity. Currently, that excess capacity is used to process ore for Bre$-$Y Mining Inc., in exchange for a payment from Bre$-$Y of $$400,000$ per year. The out$-$of$-$pocket cost of running the furnaces $($which would be unchanged in any case$)$ is $$1\mathrm{.}4$ million per year, and processing the gold from Tomkat$$s mine would require $40\%$ of the furnace capacity.

The discount rate is $10\%$ per year. You can assume that TomKat has several million dollars of taxable income from other profitable operations.

$($a$)$ Forecast the incremental cash flows to TomKat that would result from reopening the mine, on a year$-$by$-$year basis.

$($b$)$ Compute the NPV and the IRR of the proposal to reopen the gold mine at this point in time.