A new home was purchased in Niagara Fall$$s Butler Heights subdivision for $$375,000$ by Karen. The house is going to be financed by making a down payment of $15\%$ and financing the balance with a conventional mortgage. The mortgage is to be amortized over $20$ years with weekly payments. The rate chosen for the first four years of the mortgage is $2\mathrm{.}79\%$ compounded semi$-$annually. Show all work with formulas

a$)$ Determine the size of the weekly payments?

b$)$ What will the outstanding balance be after the first four years?

c$)$ Determine the size of the new weekly payment if after the first four year term she renews the mortgage using a fixed rate of $2\mathrm{.}04\%$ compounded semi$-$annually.