"It's my birthday soon, let's go celebrate together!" => 곧 내 생일이니까 같이 (나가자 / 식사하자 / 자축하자 ). 자축하다 means "celebrated oneself" (자=self, 축=celebrate), so 자축하자 is "Let's celebrate ourselves". But it sounds a little too formal to use in this context. Most people would just say 나가자(Let's go out) or 식사하자(Let' have a nice dinner) since going out already implies a celebration. If it's for someone else, 축하해 주자 (give someone a celebration) might work well too.

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