Ms$.$ Esperanto obtained a $$40,600$ home equity loan at $10\%$ compounded monthly. $($Do not round intermediate calculations. Round the PMT and final answers to $2$ decimal places.$)$a$.$ What will she pay monthly if the amortization period is $15$ years?b$.$ How much of the payment made at the end of the fifth year will go towards principal and how much will go towards interest?c$.$ What will be the balance on the loan after five years?d$.$ How much interest did she pay during the fifth year?