Without any hesitation; instantly.
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To do something right away, with no delay and no planning.
When someone acts at the drop of a hat, they're ready to go at any moment.
It can show eagerness, or it can show acting without thinking.
"In the Blink of an Eye"
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Happening extremely quickly, describing the speed of an event
"Without a Second Thought"
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Do something immediately with no doubt or hesitation in the mind
"At the Drop of a Hat"
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Do something immediately without any hesitation