Aaron retired after $35$ years as a teacher in Santa Barbara, California, and moved to Nevada because the state does not levy income taxes. He retired in June and moved into his new home in Nevada on September $15,2022.$ He started receiving a $$5,000$ monthly pension from the State Teachers Retirement System on July $1$st $($monthly payments are paid on the first day of each month$).$ How much of Aaron's pension will be taxed by California in Tax Year $2022?$