$($The correct answer is $0\mathrm{.}97$ months, please tell me how with all steps and formulas used$)$ Consider Lisa Simpson's banana factory. All harvested bananas are processed at the factory, which has an unlimited$-$size container that stores unprocessed bananas.

Assume the container is empty at the beginning of the year and has three seasons.

The harvest yields $6,000$ tons$/$month during the first third of the year, $8,500$ tons$/$month during the middle third of the year, and only $2,000$ tons$/$month during the final third of the year.

The factory's processing capacity equals $5,500$ tons$/$month$.$

What is the average waiting time of bananas in the container in months $($rounded to two decimal places$)?$