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Part $1$

A$$ variable$-$rate mortgage of $$$$135$ comma $000$ is amortized over $25$ years by equal monthly payments. After $12$ months the original interest rate of $8\%$ compounded semi dash annually was raised to $8\mathrm{.}5\%$ compounded semi dash annually. Two years after the mortgage was taken$$ out, it was renewed at the request of the mortgagor at a fixed rate of $8\mathrm{.}4\%$ compounded semi dash annually for a$$ four$-$year term.

$($a$)$ Calculate the mortgage balance after $12$ months.

$($b$)$ Compute the size of the new monthly payment at the $8\mathrm{.}5\%$ rate of interest.

$($c$)$ Determine the mortgage balance at the end of the$$ four$-$year term.