Pierre, a cash basis, unmarried taxpayer, had $$4,370$ of state income tax withheld during $2023.$ Also in $2023,$ Pierre paid $$1,093$ that was due when he filed his $2022$ state income tax return and made estimated payments of $$3,059$ towards his $2023$ state income tax liability. When Pierre files his $2023$ Federal income tax return in April $2024,$ he elects to itemize deductions, which amount to $$15,900,$ 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?

$fill in the blank $1$

b$.$ As a result of overpaying his $2023$ state income tax, Pierre receives a refund of $$1,748$ early in $2024.$ The standard deduction for single taxpayers for $2023$ was $$13,850.$ How much of the $$1,748$ will Pierre include in his $2024$ gross income?