Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
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Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers.
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