Interest for the initial four$-$year term of a $\backslash (\backslash$$ $105,000\backslash )$ mortgage is $\backslash (8\mathrm{.}31\backslash \%\backslash )$ compounded semi$-$annually. The mortgage is to be repaid by equal monthly payments over $15$ years. The mortgage contract permits lump$-$sum payments at each anniversary date up to $10\backslash \%$ of the original principal. $($a$)$ What is the balance at the end of the four$-$year term if a lump$-$sum payment of $\backslash (\backslash$$ $5,000\backslash )$ is made at the end of the third year? $($b$)$ How many more payments will be required after the four$-$year term if there is no change in the interest rate? $($c$)$ What is the difference in the cost of the mortgage if no lump$-$sum payment is made? $($a$)$ The outstanding balance at the end of fourth year is $\backslash$$ $.($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.$)($b$)$ The number of payments is $($Round the final answer up to the nearest whole number. Round all intermediate values to six decimal places as needed.$)($c$)$ The difference in cost is $\backslash (\backslash$$ $\backslash )($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.$)$