Peeta says that he doesn't want to lose himself once the games begin and that he wants to prove he's not just a "piece in their games." what does he mean by this?

He doesn't want to be a toy the capitol uses for entertainment. With all the horrors he is about to go through, he knows he is going to change but he doesn't want to forget who he always was. He is a good person, and in the games good people are remodeled into killers. This is Peeta refusing to conform to the goverment's will.

It should be noted that the central idea is that the foods offered in American supermarkets are very convenient but unhealthy.

A central idea simply means the theme or the main idea that's in a story. It's simply the reason why the author writes the literary work.

In this case, the main idea is that the foods offered in American supermarkets are very convenient, but unhealthy. It was stated that canned foods are unhealthy and can lead to diseases.

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