A woman deposits $$13,000$ at the end of each year for $7$ years in an investment account with a guaranteed interest rate of $6\%$ compounded annually.

$($a$)$ Find the value in the account at the end of the $7$ years.

$($b$)$ Her sister works for an investment firm that pays $5\%$ compounded annually. If the woman deposits money with this firm instead of the one in part $($a$),$ how much will she have in her account at the end of $7$ years?

$($c$)$ How much would she lose or gain over $7$ years by investing in her sister's firm?

$($a$)$ The woman's deposits form an because the deposits are made at the of each period. Therefore, the formula should be used.