A lathe used for the purpose of cutting molded plastics was acquired $10$ years ago for a total expenditure of RM$7,500,000.$ At the time of acquisition, the machine was projected to have a lifespan of $15$ years, and the management's initial estimation, which remains unchanged, posited that the residual value would be negligible upon the completion of the aforementioned $15-$year period. The machine is subject to depreciation using the straight$-$line method, resulting in an annual depreciation expense of RM$500,000.$ As a result, the current book value of the equipment stands at RM$2,500,000.$

According to the report provided by the R&D manager, there is a proposition to acquire a new specialized machine at a cost of RM$12,000,000,$ which includes expenses for freight and installation. This machine is projected to have a lifespan of five years and is expected to result in a reduction in both labor and raw material consumption. Consequently, the annual operating costs are anticipated to decrease from RM$9,000,000$ to RM$4,000,000.$ The decrease in expenses will result in an increase in pre$-$tax profits of RM$9,000,000-$ RM$4,000,000=$ RM$5,000,000$ annually.

According to estimates, the projected salvage value of the new equipment after a period of five years is RM$2,000,000.$ The present market worth of the antiquated machine is RM$1,000,000,$ which is lower than its book value of RM$2,500,000.$ In the event of acquiring the new machine, the decision has been made to sell the old lathe to another company, rather than opting for an exchange with the new machine. The marginal tax rate of the corporation is $40$ percent, while the replacement project is characterized by a risk level slightly below the average. The net operating working capital requirements will experience an increase of RM$1,000,000$ upon replacement. The newly acquired machine is classified under the $3-$year Modified Accelerated Cost Recovery System $($MACRS$)$ class $(33\%$; $45\%$; $15\%$; $7\%).$ Additionally, due to the relatively assured nature of the cash flows associated with the project, the project's cost of capital is determined to be $11$ percent, which is lower than the $12$ percent typically assigned to projects with average risk.

a$.$ Using the Payback Period Method and Discounted Payback Period Method, calculate the number of years needed to recover the initial cash outlay.