「〜ないで」 means 'without doing ~' or can be used for a negative command 'don't do ~'. It's formed by adding 「で」 to the negative form (ない-form) of a verb. It indicates that the main action happens while the preceding action is avoided.
「〜ないで」は「〜をしないで」という意味。前の行動をせずに次の行動をするときに使う。
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