You are an analyst working for Goldman$$ Sachs, and you are trying to value the growth potential of a$$ large, established$$ company, Big Industries. Big Industries has a thriving$$ R&D division that has consistently turned out successful products. You estimate$$ that, on$$ average, the division launches two projects every three$$ years, so you estimate that there is a $69\%$ chance that a project will be produced every year.$$ Typically, the investment opportunities the$$ R&D division produces require an initial investment of $10\mathrm{.}5$ million and yield profits of $0\mathrm{.}97$ million per year that grow at one of three possible growth rates in$$ perpetuity: $3\mathrm{.}2,0,$ and $-3\mathrm{.}2.$ All three growth rates are equally likely for any given project. These opportunities are always$$ "take it or leave$$ it$"$ opportunities: If they are not undertaken$$ immediately, they disappear forever. Assume that all the probabilities are$$ risk$-$neutral probabilities, which means the cost of capital is always the$$ risk$-$free rate and$$ risk$-$free rates follow this$$ path: The current interest rate for a$$ risk$-$free perpetuity is $7\mathrm{.}7\%$; in one$$ year, there is a $65\%$ chance that all$$ risk$-$free interest rates will be $9\mathrm{.}7\%$and stay there forever and a $35\%$ chance that they will be $5\mathrm{.}7\%$ and stay there forever. The current$$ one$-$year risk$-$free rate is $6\mathrm{.}8\%.$ What is the present value of all future growth opportunities Big Industries will$$ produce?$($Hint: Make sure to round all intermediate calculations to at least four decimal places.$)$