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Hello. I would really like you to help me understand whether the object in a narrative sentence is always needed or not. For example:

-As far as she figured (it) out after coming to the school, the item allowed to get into a person's mind.

Do I need to put "it" in this sentence? Would the sentence get worse if I took "it" out of it? Could you please explain in what context the omission of a dummy object is admissible, and in what it is not. Thanks in advance!

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"figure out" is normally transitive. We do not say "I don't understand how this works. I can't figure out." We say "I don't understand how this works. I can't figure it out." Even so, "As far as she figured out" seems loosely possible to me. At least, it does not seem glaringly impossible. I'm not really sure why this is. It may be through some association of ideas such as "She figured out that much" -> "She figured out that far" -> "As far as she figured out". Interestingly, "As far as she could figure out" seems even more unobjectionable to me.

Derevenshinathe item allowed to get into a person's mind

This does not make sense (though it does not affect your main question). Possibly "allowed one to get" is meant.

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Thanks for making the first one clear, but why is "allow to do" without an object wrong?
DerevenshinaThanks for making the first one clear, but why is "allow to do" without an object wrong?

It is just the nature of the verb "allow" that it needs an object (except in special expressions or with a clause). Something cannot "allow to ~". It has to "allow someone/something to ~".