$1.$ A hospital director is considering two alternative investment programs. Both have costs of $$5,000$ in year $1$ only. Project $1$ provides benefits of $$2,000$ in each of the first $4$ years only. Project $2$ provides benefits of $$2,000$ for years $6$ to $10$ only.

a$.$ Compute the net benefits using a discount rate of $6$ percent.

b$.$ Knowing that the calculations are dependent on the discount rate, conduct a sensitivity analysis by re$-$calculating with a discount rate of $12$ percent. Based on your calculations, which investment program should the director choose?