Cat and Claire are having a conversation about Cat’s favorite number. Cat says, "My favorite number is a two-digit perfect square!"

Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?"

Cat says, "Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number."

Claire says, "Now I know your favorite number!" What is Cat’s favorite number?

Let's think step by step and output the final answer within \boxed{}.
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