Lucia $(40)$ is single. Her $2022$ federal itemized deductions totaled $$17,400,$ of which she claimed $$1,200$ of state income taxes paid in $2022.$ She filed her tax return on April $14,2023,$ and on May $1,2023,$ she received a $$1,200$ state income tax refund. How much of her $2022$ state income tax refund is taxable on her $2023$ federal return?

$$0$

$$500$

$$1,200$

$$2,000$

$$0$

$$250$

$$1,000$

$$2,500$