On January $1,2025,$ Rich Inc. loans $$514,404$ to Poor Co$.$ in exchange for a $$600,000$ zero$-$interest$-$bearing promissory note payable in $2$ years, with a maturity date of $\frac{1}{1}\frac{?}{2027}.$ The market rate of interest for a transaction of this nature for Poor Co$.$ is $8\%.$ Rich Inc. has a calendar year$-$end, and it uses the effective$-$interest method to record the lending transaction.

$($round to the nearest dollar$)$

$(1)$ Write a journal entry on $\frac{1}{1}\frac{?}{2025}$ for Rich Inc. to recognize the loan $($show calculations$).$

$(2)$ Write a journal entry on $\frac{12}{31}\frac{?}{2025}$ for Rich Inc. to recognize the amortization on the Discount on Notes Receivable and the interest earned in $2025($show calculations$).$

$(3)$ What is the carrying value of the promissory note as of $\frac{12}{31}\frac{?}{2025}?($calculation only, no journal entry$).$

$(4)$ Write a journal entry on $12/31/2026$ for Rich Inc, to recognize the amortization on the Discount on Notes Receivable and the interest earned during $2026($show calculations$).$

$(5)$ Write a journal entry on $\frac{1}{1}\frac{?}{2027}$ for Rich Inc. to recognize the receipt of the repayment of the note.