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$Alice$ and $Bob$ both choosed a secret number and secretly tells the numbers to $Charlie$. Then they had the following conversations between them (which is not secret) :

• $Charlie$ : Both of you have choosen a positive integer and one of them is $2$ more than the other.
• $Alice$ : I don't know $Bob$'s number.
• $Bob$ : I don't know your number either. Do you know now?
• $Alice$ : I still have no idea. Do you?
• $Bob$ : Yes, I know now.
• $Alice$ : Then so do I.

Find all possible secret number choosen by $Alice$.