Your friend tells you he has a very simple trick for taking one$-$third of the time it takes to repay your mortgage: Use your Christmas bonus to make an extra payment on January $1$ of each year $($that is$,$ pay your monthly payment due on that day twice$).$ Assume that the mortgage has an original term of $30$ years and has an APR of $12\%.$ a$.$ If you take out your mortgage on January $1($so that your first payment is due on February $1),$ and you make your first extra payment at the end of the first year, in what year will you finish repaying your mortgage? b$.$ If you take out your mortgage on July $1($so that the first payment is on August $1),$ and you make the extra payment each January, in how many months will you pay off your mortgage? c$.$ How will the amount of time it takes to pay off the loan given this strategy vary with the interest rate on the loan?