Assume that you are $20$ years old but decide to wait before saving for retirement. You decide to start saving later when you are $45$ years old. As a result, you start saving on January $1,2047.$ You plan to retire on December $31,2071,$ when you are $70$ years old. There are $25$ years from the time you started investing $($saving$)$ until you retire. When you start investing in $2047,$ you have no previous or other retirement savings. Assume there are $365$ days and $52$ weeks in each year from $2047$ to $2071.($Ignore leap years$).$ Assume that taxes will not affect any of the amounts or your savings.

You invest $$400$ at the end of each month into a retirement account paying $7\%$ compounded monthly for $14$ years starting on January $1,2047.$ After $14$ years, you do not make any more payments or withdrawals and leave the money in the retirement account until retirement. Show all work and answer the following questions:

$1.$ Assuming no withdrawals or additional payments were made, how much money will be in your retirement account after $14$ years?

$2.$ After $14$ years, how many years are left until you retire?

$3.$ Assuming you made all the monthly payments for $14$ years and left the money in the account without making any additional payments or withdrawals, how much money will be in your retirement account when you retire after $25$ years?

$4.$ Assuming you made all the monthly payments for $14$ years and left the money in the account without making any additional payments or withdrawals, how much did you pay into your retirement account over the $25$ years?

$5.$ Assuming you made all the monthly payments for $14$ years and left the money in the account without making any additional payments or withdrawals, how much interest did you earn over the $25$ years?