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Pincher Attack, Discovering the Island
On Dolce Island, there's a cute little Pichu playing a tune on their ukulele for its friends. The same enemies also reach here, and using something they refer to as a Control Gauntlet, they start to control the Pokémon nearby.
You learn from Professor Hastings that this group is called The Pokémon Pinchers, and they've been stealing Pokémon left and right.
You wash up on the shore of Dolce Island, but your Styler is malfunctioning now. Reprogram it by entering your name, and head north.
You'll run into the Pichu you saw earlier, and it seems a bit upset. Befriend it through a capture, remembering to avoid its attacks and filling its Friendship Gauge.
Pichu will play, and an old local man named Booker will introduce himself to you. He invites you to Renbow Island, so head north with him to reach his boat on the Eastern Beach.
Now that you've met Booker, the local Pokémon start to relax, and give you a chance to capture them. Take this opportunity to befriend more Pokémon and fill your Browser.
When you see the Sentret, head east. A log is blocking the path, but the friend Pokémon you have made may be able to help you. Perform a Target Clear, using a Pokémon with the "Cut 1" field move. Pokémon nearby that have that field move are Sunkern, Krabby, Bulbasaur and Wingull.
After finally making it to Eastern Beach, you must now push Booker's boat down to the water. Find a Pokémon to help you with this Target Clear. The nearby Ursaring would be a perfect friend.
Talk with Booker and say yes to make your way to Renbow Island. The Pichu that was befriended would be here too.
And now off to Renbow Island