A $30-$year$-$old student wants to save $$10$ a day for her retirement. Every day she places $$10$ in a drawer. At the end of each year, she invests the accumulated savings $)$ in a brokerage account with an expected annual return of $5\%.$

$(1)$ If she keeps saving in this manner, how much will she have accumulated at age $70?$

$(2)$ If a $45-$year$-$old investor began saving in this manner, how much would he have at age $70?$

$(3)$ How much would the $45-$year$-$old investor have to save each year to accumulate the same amount at $70$ as the $30-$year$-$old investor?