Thank you very much
Great!! I was hoping that one might appeal to you.
I am not sure about the details of the passport, but I think most of the citizens of the Kingdom have it if they go to Ba Sing Se. If Mei does not have one, then she could borrow it from one of Shen's best friends Zyn, and maybe bluff her way through. Another option is to go around the Serpent's Pass which was guarded by a monster 500 years ago (I don't know about now though).
Route One (rough plan):
Day 1: Cut through the mountains north to Shen's hideout (one of three) between the southeastern edge of the desert and the mountains. There, three ugly bikes can be found in the hideout. It was put together by Shen's second friend (who is a genius self-taught mechanic). There are also rags to protect from the hot sun. Rest
Day 2: Ride parallel to the mountains where they can occasionally take shelter inside some caves. Reach the second hideout in the mountains just outside the northeastern part of the desert.
Day 3: Ride to 32, Full Moon Bay (sightseeing possible). Bury the bikes with a large size stone that Shen can find later. Take Ferry or the dangerous Serpent's Pass (33) which will add a day.
Day 3 ~ 4: 16 hours of walking to the outer wall of the Ba Sing Se. Early parting with a gift.
Day 5: City experience. Late parting with a gift.
Route Two (rough plan):
Night 1
28 (Shen's hometown): Where Mei could be invited to spend a comfortable night at Shen's family's home. Many of Ai's beautiful clay artwork will be displayed. Food and Chat.
Day 1 ~ 2
27 (Plain's Village): A chaotic town with little economical development.
24 (Foggy Swamp): Home to the swamp benders (is that what they are called?). I remember seeing that episode. Is Mei interested in learning a bit about that type of bending?
31 (Misty Palm Oasis): A beautiful city on the southern edge of the desert (more towards the west side). The natural wonder of the glacier at the center of the city has returned. It is a great resting spot filled with fun activities (details of which I have not decided), good food, and drinks. Shen might be able to tell a legend of Toph here.
Day 3 ~ 6
31 to 32 Skirting the edge of the desert. They can learn more about each other. It is during this time that Shen will make the pendant. The desert heat will harden the clay rather than using a kiln. There are a lot of distances to cover which many events can happen or some interesting locations can be added. Maybe Shen can build a small model of what he thinks the buried library of the desert looks like.
Day 7 ~ 8 Full Moon Bay or Serpent's Pass (extra day). Reach Ba Sing Se. Gift.
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How is it?
Those are kind of rough ideas that I am listing out. There are a lot of unexplored potentials here, but I think I am ready to roleplay it out. Warning: I am a slow writer.
Mei can bring home a lot of stories for her grandmother.