You buy a house and borrow $$500,000$ on a $"7/23$ balloon" mortgage with an APR of $6\%$ with monthly compounding.

The $7/23$ balloon is set up with a monthly mortgage payment like a standard $30-$year mortgage. $(7+23=30).$

Then, if the homeowner is still in the house at the end of the $7$th year $($after the $7*12=84$th monthly payment$),$ the homeowner must pay off the remainder of the loan. This is what is called the "balloon payment".

If there are $360-84=276$ payments remaining, what will the balloon payment on this mortgage be$?$

Group of answer choices

$448,196\mathrm{.}93$

$387,121\mathrm{.}64$

$431,605\mathrm{.}96$

$406,235\mathrm{.}48$

$399,509\mathrm{.}99$