To finance a vacation in $2$ years, Elsie saves $$380$ at the beginning of every six months in an account paying interest at $13\%$ compounded semi$-$annually.

$($a$)$ What will be the balance in her account when she takes the vacation?

$($b$)$ How much of the balance will be interest?

$($c$)$ If she waits an additional year to start her vacation, and continues to save the same amount of money, how much more money does she have to spend?