A man takes out a loan of $$35000$ to be repaid by monthly instalments over $5$ years. The first payment will be made at the end of the first month. Take the interest rate to be a $12\%$ nominal rate convertible monthly.

$(1)$ Calculate the monthly instalment.

$(2)$ Suppose the man asks for the term of the loan to be extended by $1$ year after making the $12$th payment. Find the outstanding loan after the $12$th payment and hence, calculate the new instalment to the nearest dollar