Your sister turned $30$ today, and she is planning to save $$5,000$ per year for retirement, with the first deposit made one year from today and the last deposit made on her $65$th birthday. She currently has no retirement savings. She will invest in a mutual fund, which she expects to provide a return of $10\%$ per year throughout her lifetime. She plans to retire $35$ years from today when she turns $65.$ She will make a total of $25$ annual withdrawals$$with her first withdrawal made on her $66$th birthday, and the last withdrawal made when she is $90.$ After the last withdrawal there will be $$250,000$ left in the account for her heirs. Under these assumptions, how much can she spend each year after she retires? Please show detailed solutions.