Your friend is celebrating her $25$ th birthday today and wants to start saving for her anticipated retirement at age $65.$ She wants to be able to withdraw $$400,000$ from her saving account on each birthday for $25$ years following her retirement; the first withdrawal will be on her $66$th birthday. Your friend intends to invest her money in a retirement account, which earns $8$ percent return per year. She wants to make an equal annual deposit on each birthday into the account for her retirement fund. Assume that the annual return on the retirement account is $8$ percent before retirement and $6$ percent after retirement.

a$)$ If she starts making these deposits on her $26$th birthday and continue to make deposits until she is $65($the last deposit will be on her $65$ th birthday and the total number of annual deposits is $40),$ what amount must she deposit annually to be able to make the desired withdrawals at retirement? $($Hint: One way to solve for this problem is to first find the value on your friend's $65$ th birthday of the $$400,000$ withdrawal per year for $25$ years after her retirement using the annual return after retirement and then find the equal annual deposit that she needs to make from her $26$th birthday to $65$ th birthday using the annual return before retirement.$)$ Ignore taxes and transaction costs for the problem.

b$)$ Suppose that whenever your friend makes the annual deposit to the retirement account, her employer will contribute $$6,000$ to the same account every year as part of the company's profit sharing plan. In addition, your friend will receive a $$100,000$ distribution from a family trust fund on her ${25}^{th}$ birthday, which she will also put into the retirement account. What amount must she deposit annually now to be able to make the desired withdrawals at retirement?