Today is Dec. $31,2020.$ Lauren is planning to retire $40$ years from today and has decided to save money to support her living for $20$ years, i$.$e$.,2061$ through $2080,$ after her retirement $($Caution: although the timeline is similar, details provided in Question $2$ do not necessarily apply here$).$ Lauren expects that although $$60,000$ is reasonable to cover her living expenses in $2061,$ the required expenses will then grow at a rate of $4\%$ each year from $2062$ to $2080.$ Assume that the interest rate is $10\%$ and will be constant over time.

Determine how much she needs to save by the end of $2060.$