Firm $1$ must decide whether to enter an industry in which firm $2$ is an incumbent. To

enter this industry, firm $1$ must choose to build either a plant with a small output

capacity $(S),$ or large output capacity $(L).$ A plant with small capacity costs $$50$ to

set up; one with large capacity cost $$175.$ In either case, the marginal cost of

production is zero. But firm $1$ can also opt to stay out $(0),$ in which case it does not

incur any type of cost$.$ Firm $2$ is able to observe firm $1\text{'}$s decision before deciding

whether to expand or not its initial small output capacity operation. Expanding $($E$)$

costs firm $2$ $$70,$ whereas not expanding $(N)$ incurs no cost for the firm. In either

case, the marginal cost of production is also zero.

The revenues under the different scenarios are given below.

If only one small firm exists, its revenue is $$80,$ the other earns zero.

If two small firms exist, each earns revenue of $$70.$

If only one large firm exists, its revenue is $$200,$ the other earns zero.

If two large firms exist, each earns revenue of $$90.$

If one small and one large firm exist, the small firm earns $$40,$ while the large one

earns $$160.$

Answer the following:

a$)$ How many pure strategies are available for firm $1?$

b$)$ How many pure strategies are available for firm $2?$

c$)$ What is the sum of the firms' profits in the subgame perfect equilibrium?