
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
(a) The numbers  $1 , 2, 4, 8, 1 6 , 32, 64, 1 28$  are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number). After this procedure has been repeated seven times, only a single number will remain. Could this number be  $97$ ?
 
(b) The numbers  $1 , 2, 22, 23 , . . . , 210$  are written on a blackboard.
We are allowed to erase any two numbers and write their difference instead (this is always a non-negative number) . After this procedure has been repeated ten times, only a single number will remain. What values could this number have? 

(A.Shapovalov)

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