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Once you've defeated Team Galactic and your party is ready, you can head to Lake Verity and cause Mesprit to appear. Flying to Twinleaf Town is the quickest way to get to the lake. You'll find the Verity Cavern entrance in the middle of the lake. You'll see Mesprit in the center of the room. It will look like the battle is about to start, and then Mesprit will run away before it begins. Professor Rowan will inform you that Mesprit is now roaming throughout Sinnoh, and that you can track it on the map. Roaming Pokémon like Mesprit will change locations every time you move between locations. This can make tracking them down difficult, but there's a trick you can use to speed it up. Walking between these two locations is the most effective way to encounter a roaming Pokémon due to the close proximity of tall grass on each side and the low level wild Pokemon. Each time you walk from one location to the other, Mesprit will move to a new location in Sinnoh. You can see its current location flashing on your Marking Map in the lower screen. Mesprit's movements are random, so you may have to go back and forth many times until you're both in the same spot. Once you've changed locations and Mesprit is in the same location you are, use a Repel so that you don't encounter any other Pokemon. If you encounter another Pokémon or start a battle with a trainer, Mesprit will move and you'll have to start over. Save before attempting to encounter Mesprit so you can quickly try again if you accidentally mess up. Since you're using a Repel, you won't encounter any Pokémon lower than your first Pokemon's level. This will prevent you from encountering any of the wild Pokémon in these two areas. If your first Pokémon is over level 50, the Repel will prevent you from encountering Mesprit.
Travel to Lake Verity. Surf to the middle of the lake. Enter Verity Cavern. Talk to Mesprit in the cavern. Travel to Route 205 or Valley Windworks. Move back and forth between the two locations to cause Mesprit to move. Use a Repel once you and Mesprit are in the same location. Save your game. Run through the tall grass or surf until you encounter Mesprit.