Five years ago, Diane secured a bank loan of $$350,000$ to help finance the purchase of a loft in the San Francisco Bay area. The term of the mortgage was $30$ years, and the interest rate was $6\%/$year compounded monthly on the unpaid balance. Because the interest rate for a conventional $30-$year home mortgage has now dropped to $2\mathrm{.}5\%/$year compounded monthly, Diane is thinking of refinancing her property. $($Round your answers to the nearest cent.$)$ Her current payment is $$2098\mathrm{.}43$ a month.

$($a$)$ what is her current outstanding principal?

If Diane decides to refinance her property by securing a $30-$year home mortgage loan in the amount of the current outstanding principal at the prevailing interest rate of $2\mathrm{.}5\%/$year compounded monthly,

$($b$)$what will be her monthly mortgage payment?

$($c$)$ How much less will Diane's monthly mortgage payment be if she refinances?