A mechanical engineer deposits $$3,000$ at the end of each quarter into an account earning $6\%$ compounded quarterly. Starting at the end of the fifth deposit, she increases each subsequent deposit by $$200.($a$)$ Determine the account balance after $10$ years. $($b$)$ If instead she waits until the end of each year to make four equal annual deposits that produce the same final balance, what should each annual deposit be$?$