I'm at the end (I presume) of the 6th level, in some kind of room with a lava cross in the middle.

On the left, there's a machine that will help me go to the other side. Now, the pulley part is missing but I got it. But I can't figure out how to use it. I've tried from multiple locations, multiple times, but I can't seem to find a way to replace this part. Any suggestions?

Push the small box and climb on it. Now you can fix the pulley...

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cheers