From the Super Mario Wiki
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After activating the Mini-Mario, the player has to use a Falling Spike to reach a set of stairs they both need to climb, using the ladder at the start to move the Boo out of the way. Once at the top of the stairs, the player can use the Mini-Mario as a step or do a Handstand Jump to reach a platform with a blue Color Switch. The player has to press the Color Switch and have the Mini-Mario drop onto the platforms below, then the player needs to press a red Color Switch when the Mini-Mario is on the blue stair platforms. With the Mini-Mario on the lower platform, the player has to climb down a ladder to reach the locked door.