$6.$ AGE Inc.$$s assets in place will be worth either $$250$ million or $$100$ million in one year depending on the state of the business conditions. The good and bad states have $60\%$ and $40\%$ probabilities, respectively. The firm has $$150$ million $($face value$)$ outstanding debt that is due next year. AGE Inc. has an opportunity that requires an investment of $$50$ million and offers a safe return of $$75$ million in one year. The current risk$-$free rate is $15\%.$

a$.$ Should the firm undertake the project?

b$.$ Can the firm successfully raise new equity from its shareholders to fund this investment opportunity? If not, what are the options of the managers for convincing the shareholders to fund the project? Please explain.

c$.$ Would your answers to the previous question if the face value of the debt were $$100$ million? Please explain