Six Face Die

Roll a fair 6-sided die six times and record the result of each roll. You then write these numbers on the six faces of another, unlabeled fair die.

For example, if your six rolls were:

3, 5, 3, 6, 1, 2

Then your second die wouldn’t have a 4 on it; instead, it would have two 3s.

Next, you roll this second die six times. You take those six numbers and write them on the faces of yet another fair die, and you continue this process of generating a new die from the previous one.
Eventually, you’ll have a die with the same number on all six faces.

How would you go about finding the average number of rolls it will take to reach this state?