Using a $$400,000,30-$year monthly payment Priced Level$-$Adjustable Mortgage, assuming the "real" loan rate is $5\%,$ with inflation rates of $5\%,6\%,$ and $7\%$ for years $1,2,$ and $3,$ respectively. It is assumed that adjustments are made annually in the outstanding balance. What is the unpaid mortgage at the beginning of the $4$ th year?

$$454,182\mathrm{.}56$

$$461,361\mathrm{.}27$

None of the possible numerical answers given is correct

$$434,727\mathrm{.}96$

$$444,469\mathrm{.}68$