A couple, during their honeymoon, when they are $25$ years old, decide to save some money by reducing unhealthy food, $$12$ per day, $$360$ per month. They plan to deposit their monthly saving money $$360$ into a S&P $500$ Indexed Mutual Fund Account which returns $6\%($in average$)$ annually that is $0\mathrm{.}5\%$ monthly. Let their monthly account balance to be $B_n,B_0=$$$360.$

Write the first four terms of the sequence $\{B_n\}.$

Find the recurrence relation that generates the sequence.

Determine their account balance at their retirement time, $70$ years old, $B_{540}.$

They are planning to keep the account and start withdrawing $$5,000$ per month, starting the next month after their retirement celebration. Determine for how long their saving in this account will last throughout their retirement.