How would you say "I'll do it after I finish drinking this cup of coffee"
Context: this is part of a dialogue where the reader understands that the action being referenced here is to finish writing a letter to their mother.

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等我喝完这杯咖啡就去做

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等我喝完这杯咖啡就去做

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“我喝完就去”
“我喝完这杯咖啡就去开始做”

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I would say:
我喝完这杯咖啡后再去写。
(I'll do it after I finish drinking this cup of coffee)

ps: Chinese don't use 做 to describe writing a letter. In English, we can make a reference to an action by saying "do it". In Korean, we have the "하다". But in Chinese, we can't use "做" in that manner.

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Since you have written the context,  I would say：
我喝完这杯咖啡就去写信。