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コートを着なければいけません。

I'm not sure about the meaning of this sentence. Would it be something like "If you don't wear a coat, you're not going"?

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なければいけません means "must do".

So the translation becomes:
コートを着なければいけません。
You have to wear a coat.

なければいけません is one of the variants of なければならない。

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