An employee plans to invest in a retirement fund at the beginning of each of the next $12$ years.

The employee will initially invest $$50,000$ in the first year and increase the annual investment

amount by $3\%$ in each subsequent year. The employee believes she will earn $10\%$ annually on

her investments in the first $3$ years, then $8\%$ annually in the next $2$ years, then $6\%$ annually in

the next $4$ years, and then $5\%$ annually in each of the final $3$ years before she retires.

a$.$ How much money will the employee have in the retirement fund when she retires?

b$.$ What would be the internal rate of return associated with her investments and her final

retirement position?