Peter deposits $$500$ into a fund at the end of each year for $5$ years. At the end of $10$ years, he makes an additional deposit $X$.$ At the end of $20$ years, Peter uses the accumulated balance in the fund to buy a $10-$year annuity$-$immediate with annual payments of $$1000$ per year. Assume the annual effective rate of interest is $5\%.$a$)$ At the end of $20$ years, calculate the present value of the $10-$year annuity$-$immediate with annual payments of $$1000$ per year;b$)$ Calculate the value of $X$.$