Pierre, a cash basis, unmarried taxpayer, had $$4,820$ of state income tax withheld during $2022.$ Also in $2022,$ Pierre paid $$1,205$ that was due when he filed his $2021$ state income tax return and made estimated payments of $$3,374$ towards his $2022$ state income tax liability. When Pierre files his $2022$ Federal income tax return in April $2023,$ he elects to itemize deductions, which amount to $$14,800,$ including the state income tax payments and withholdings, all of which reduce his taxable income.

a$.$ What is Pierre's $2022$ state income tax deduction?

b$.$ As a result of overpaying his $2022$ state income tax, Pierre receives a refund of $$1,928$ early in $2023.$ The standard deduction for single taxpayers for $2022$ was $$12,950.$ How much of the $$1,928$ will Pierre include in his $2023$ gross income?