A young executive is going to purchase a vacation property for investment purposes. She needs to borrow $$106,000\mathrm{.}00$ for $29$ years at $4\mathrm{.}2\%$ compounded monthly, and will make monthly payments of $$527\mathrm{.}33.($Round all answers to $2$ decimal places.$)$

a$)$ What is the unpaid balance after $13$ months?

b$)$ During this time period, how much interest did she pay?