Now that you finally have the correct type of crystal, it is time to take it to Cid.
This quest is available if you have completed the quest The Curious Case of Giggity.
Starting the Quest
Talk to Hedyn near Mih Khetto's Amphitheatre in Gridania to begin the quest.
Deliver the Corrupted Crystal to Cid
Use an aetheryte to go to the Airship Landing. Talk to Lionnellais (the Airship Ticketer) to go to the area where Cid is waiting. Turn right to go outside, and go along the wooden walkway to reach Cid. Talk to him to hand over the corrupted crystal and complete the quest. There will be a cutscene.
Next Main Scenario Quest
After you complete the quest, you will need to be at least level 44 to proceed to the next quest. After you reach level 44, talk to Cid to accept the next quest: Lady of the Vortex. There will be a cutscene. This one is important, so it is recommended to watch it either now or via the Unending Journey in an inn room.