If Jack invests $$1$ in investment A at the beginning of year $1,$ he will receive $$0\mathrm{.}50$ at the beginning of year $2$ and another $$0\mathrm{.}80$ at the beginning of year $3.$ Alternatively, he can invest $$1$ in investment $B$ at the beginning of year $2$ and receive $$1\mathrm{.}25$ at the beginning of year $4.$ Entries of $""$ in the table indicate times when no cash inflows or outflows can occur. At the beginning of any year, Jack can place money in a money market account that is expected to yield $8\%$ per year. He wants to keep at least $$50,000$ in the money market account all times and doesn't want to place any more than $$500,000$ in any single investment. How would you advise Jack to invest his winnings if he wants to maximize the amount of money he'll have at the beginning of year $4?$

a$.$ Formulate an LP model for this problem.

b$.$ Create a spreadsheet model for this problem and solve it using Solver.

c$.$ What is the optimal solution?