Lucy currently purchased a home with a $30-$year fixed rate mortgage with the loan amount of $$930,000$ at an annual rate of $5\mathrm{.}25\%$ in early May $2024.$ The refinance cost is $$5,000$ and she is watching the interest rates in order to refinance when the rate is lower.

a$.$ What is the highest annual rate that can start her to refinance after one year if she needs her future monthly payment be at least $400$ lower than before? $5$

b$.$ What is the highest annual rate that can start her to refinance after one year if she can take an annual effective cost of at most $5\%$ on the refinancing? Lucy plans to hold the property till the end of September $2030.$ Hint: this rate is higher than part a$.5^{?}$

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