$2.$ You have recently been hired as a consultant for a personal financial planning firm. One of your first projects is creating a retirement plan for a young couple, Paul and Page Lindsay. They have just celebrated their $35$th birthdays and have decided to get serious about saving for retirement.

Paul and Page hope to retire $30$ years from now $($on their $65$th birthdays$),$ and they expect to live until age $85.$ Their hope is to be able to withdraw $$140,000$ a year from their retirement account $$ the first withdrawal will occur on their $65$th birthdays, and the $20$th and final withdrawal will occur on their $84$th birthdays. They expect to leave their children a total inheritance of $$800,000$ to split amongst themselves. So$,$ on their $85$th birthdays, the account is expected to have an $$800,000$ value $($i$.$e$.,$ they expect to leave their children a total inheritance of $$800,000).$

Paul and Page currently have $$25,000$ saved in a retirement account, which consists of a portfolio of mutual funds that is expected to produce an annual return of $7\%.$ To accomplish their goals, they would like to deposit an equal annual amount into their account starting one year from today $($on their $36$th birthdays$)$ and continue to make those deposits through age $65.($Again$,$ the account has an expected annual return of $7\%.)$ Thus, they will make $30$ annual end$-$of$-$year deposits to this account.

a$.$ How much do Paul and Page need to deposit into the account at the end of each of the next $30$ years to accomplish their goals