Please show your work and explain your answers. $($please make sure the answer is correct chat gp sometimes has errors$)$

$1.$ Sanford can choose either a safe or a risky project. To keep things simple, let each project cost $$100.$ A safe project yields $$114$ with certainty, while a risky project is equally likely to yield $$210$ or zero. Sanford needs financing for $50$ percent of the cost of his project. Lenders cannot observe his choice of project. Everyone is risk neutral, and the risk$-$free rate is $6\%.$

a$.$ If Sanford were to sell $$50$ worth of bonds with face value equal to $$53,$ in which project would he invest? Justify your answer. How much would bondholders get paid, on average?

b$.$ How much face value would Sanford need to offer lenders in order to sell $$50$ worth of bonds?

c$.$ Is Sanford willing to offer the face value required to sell bonds? Explain.

d$.$ Is Sanford willing to finance a project by selling shares? Are savers willing to purchase his shares? Does financing with equity yield an efficient equilibrium?