If I change "stay" to "keep" here, does the meaning stay the same?
Yes the meaning would be the same. This is one of the context where the two are interchangeable. "Stay" of course means to remain or not change and, in this context "Keep" can be used to mean the same thing. Ex: You need to stay calm. Ex: You need to keep calm. Ex: It's my job to stay alert. Ex: It's my job to keep alert.
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