Professor$$s Annuity Corp. offers a lifetime annuity to retiring professors. For a payment of $$83,000$ at age $65,$ the firm will pay the retiring professor $$675$ a month until death.

a$.$ If the professor$$s remaining life expectancy is $20$ years, what is the monthly interest rate on this annuity? $($Do not round intermediate calculations. Enter your answer as a percent rounded to $2$ decimal places.$)$

b$.$ What is the effective annual interest rate? $($Do not round intermediate calculations. Round your answer to $2$ decimal places.$)$

c$.$ If the monthly interest rate is $0\mathrm{.}50\%,$ what monthly annuity payment can the firm offer to the retiring professor? $($Do not round intermediate calculations. Round your answer to $2$ decimal places.$)$