Sentence Correction (SC) | Re: the usage of "for" and "to"

I knew "to" is used to express some aims or goals, and in many SC answers we replace "for" to "to". I wonder in what circumstances should we use "for".
p.s: I know "by" often appear in some wrong answers which were corrected with "to", too. If any one can compare these 3 word's usage I will appreciate it very much, thanks

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