How is the plot advanced when Nagaina started targeting the human family? 

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The part when Nag and Nagaina decide to target the human family in the house furthers the plot as part of the rising action. A few hours before, Rikki-tikki proves his quickness against the cobras when he dodges their come-from-behind sneak attack. Rikki-tikki then proves his lethality when he kills Karait. At that point in the story, the cobras know that Rikki-tikki is dangerous, and he knows that they are dangerous. An uneasy truce could have been established; however, Nag and Nagaina decide to escalate the tension. The snakes believe that if the humans are eliminated, Rikki-tikki will leave of his own accord or will be easier to hunt.  

"I will go, but there is no need that we should hunt for Rikki-tikki afterward. I will kill the big man and his wife, and the child if I can, and come away quietly. Then the bungalow will be empty, and Rikki-tikki will go."

Rikki-tikki overhears the conversation between Nag and Nagaina. Rikki-tikki feels it is his duty to protect the family, so he knows that he must kill Nag to eliminate the threat. That causes a further escalation in tensions because Nagaina wants vengeance. The plot continues to rise until the climax of the story when Rikki-tikki follows Nagaina down into her snake hole.

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