$($ please answer with clear calculations and dont use excel$)$ Luckoil Inc. owns a lease to extract crude oil from sea. It is considering the construction of a deep$-$sea oil rig at a cost of $$75$ million which is expected to remain constant. The extraction costs are $$\frac{35}{b}bl.$ The quantity of oil $Q=300,000bbl$ per year forever. The cost of capital is $6\%$ per year $($ignore taxes$).$

Next year when the rig is constructed the price is expected to be either $$\frac{60}{b}bl$ or $$\frac{30}{b}bl($and is expected to remain constant$)$ with $65\%$ and $35\%$ probability, respectively. The firm can cap the rig at a cost of $$10$ million $($cost of exit$).$ The firm can restart pumping when oil price is more favorable.

a$)$ Calculate NPV of the project without the abandonment option

b$)$ Calculate NPV with the option to exit

c$)$ Calculate value of the option to exit