On October $1,$ the home mortgage balance was $$83,000$ for the home owned by Alice Williams. The interest rate for the loan is $7$ percent.

Assuming that Alice makes the October monthly mortgage payment of $$664,$ calculate the following:

$($a$)$ The amount of interest included in the October payment $($round your answer to the nearest cent$).$

$($b$)$ The amount of the monthly mortgage payment that will be used to reduce the principal balance.

$($c$)$ The new balance after Alice makes this monthly mortgage payment.

$($a$)$ Interest amount:

$($b$)$ Principal reduction:

$($c$)$ New balance: