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The two ways to complete this level are:

[edit] Race to the exit

Swim all the way through the level, avoiding any fish that might try to touch you. You can swim up and over everything (go outside of the screen's boundary) until the green tube and then decide if you want to take the normal or secret exit. Enter the green tube at the very end to complete the level. It's as easy as that!

In order to proceed on the Star Road, you will need to find the lock and key.

[edit] Secret exit

Instead of entering the green tube though, swim along the path underneath it. This is much easier if you've got a Yoshi who can eat up all of the fish that would normally chase you. At the end of the path, you will find the key and the lock!