
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
The 97 numbers \( \frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97} \) are written on a blackboard. We repeatedly pick two numbers \( a, b \) on the board and replace them by \( 2ab - a - b + 1 \) until only one number remains. What are the possible values of the final number?

Present the answer in LaTex format: \boxed{Your answer}
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