Pierre, a cash basis, unmarried taxpayer, had $$4,250$ of state income tax withheld during $2023.$ Also in $2023,$ Pierre paid $$1,063$ that was due when he filed his $2022$ state income tax return and made estimated payments of $$2,975$ towards his $2023$ state income tax liability. When Pierre files his $2023$ Federal income tax return in April $2024,$ he elects to itemize deductions, which amount to $$15,450,$ including the state income tax payments and withholdings, all of which reduce his taxable income.

a$.$ What is Pierre's $2023$ state income tax deduction?

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b$.$ As a result of overpaying his $2023$ state income tax, Pierre receives a refund of $$1,700$ early in $2024.$ The standard deduction for single taxpayers for $2023$ was $$13,850.$ How much of the $$1,700$ will Pierre include in his $2024$ gross income?

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