Numbers $\frac{49}{1}, \frac{49}{2}, \ldots , \frac{49}{97}$ are written on a blackboard. Each time, we can replace two numbers (like $a, b$) with $2ab-a-b+1$. After $96$ times doing that action, one number will be left on the board. Find all the possible values for that number.

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