To make a great effort, especially to be fair or helpful.
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To try extremely hard to help someone, far beyond what anyone expects.
When you bend over backwards, you put in much more effort than you need to, sometimes at a cost.
It's often used when a person is being especially kind.
"Go the Extra Mile"
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Put in more effort than required, but not to the point of strain or sacrifice
"Move Heaven and Earth"
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Do everything possible no matter how difficult, focused on achieving a goal
"Bend over Backwards"
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Make an extreme effort to help or accommodate someone