When Gustavo and Serrana bought their home, they had a $5\mathrm{.}2\%$ loan with monthly payments of $$870\mathrm{.}60$ for $30$ years. After making $78$ monthly payments, they plan to refinance for an amount that includes an additional $$35,000$ to remodel their kitchen. They can refinance at $4\mathrm{.}4\%$ compounded monthly for $25$ years with refinancing costs of $$625$ included with the amount refinanced.

What is the payoff amount for their original loan after making the $78$ monthly payments? $($Round your answer to the nearest cent.$)$

$

What is the total cost of their remodel and refinancing fee? $($Round your answer to the nearest cent.$)$

$

Find the amount refinanced. $($Round your answer to the nearest cent.$)$

$

$($b$)$ Find their new monthly payment. $($Round your answer to the nearest cent.$)$

$

$($c$)$ How long will it take to pay off this new loan if they pay $$1,200$ each month? $($Round your answer up to the next whole number.$)$