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Note: this only works in the Genesis version, not the Lynx version.
Keep an eye on your computer map, and work your way to the room just east of the starting point, which is the top of a long shaft. Shoot the wall to access the foothold next to the starting point. It looks impossible, but you can perform a long jump just before the ice begins and make it across the shaft. Shoot the wall on the other side and you'll suddenly be at the exit. You can complete this mission without picking any mushrooms.