Alex is planning to take out a mortgage of $$300,000$ to purchase a home. The mortgage has a fixed interest rate of $4\mathrm{.}2\%,$ and the interest is compounded monthly. The mortgage term is $20$ years, and the amortization period is also $20$ years. Calculate the following:

a$)$ The Effective Annual Rate $($EAR$)$ for the mortgage.

b$)$ The monthly mortgage payment.

c$)$ If Alex decides to refinance the mortgage after $8$ years, calculate the new monthly payment and the adjusted Effective Annual Rate for the remaining term, considering the remaining amortization period.