On average, how many flips of a "three-sided coin" are needed to get each side at least once?

This question seems similar to an earlier question by me. That one was about a 2-sided coin (a normal coin). This question is about what would happen if I added an extra side.

You have either a fancy 3-sided coin or a fancy 3-sided die. You flip this coin until you have flipped each side at least once. What is the average (mean) number of flips required to achieve this?

For an example, the flips HTHS required 4 flips, while the sequence HTS required 3.

From the other question, I have found some things that need to be explained. The number of flips at which there is a 50–50 chance of getting all three sides is not necessarily the mean, but it is the median. I am also not asking for the minimum/maximum number of flips required.