A young married couple has carefully looked at their budget. After review, they can afford a monthly mortgage payment of $$1,172\mathrm{.}68.$ They go to their local banker and she offers them a mortgage of $5\mathrm{.}76\%$ APR with monthly compounding with a term of $30$ years. The couple has enough savings to pay $20\%$ down, so the mortgage will be $80\%$ of the home$$s value.

What is the mortgage that the couple can apply for based on their budget and the offered terms?