Suppose you start saving for retirement when you are $45$ years old. You invest $$6,000$ the first year and increase this amount by $4\%$ each year to match inflation. How much money will you have saved after $30$ years if the interest rate earned on the retirement account is $8\%$ per year?

$($a$)$ You will have saved s $($Round to the nearest dollar.$)$

$($b$)$ On the other hand, you will have saved $ $$ if the interest rate was $4\%$ instead of the original $8\%.$