
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
Amy and Bob choose numbers from  $0,1,2,\cdots,81$  in turn and Amy choose the number first. Every time the one who choose number chooses one number from the remaining numbers. When all  $82$  numbers are chosen, let  $A$  be the sum of all the numbers Amy chooses, and let  $B$  be the sum of all the numbers Bob chooses. During the process, Amy tries to make  $\gcd(A,B)$  as great as possible, and Bob tries to make  $\gcd(A,B)$  as little as possible. Suppose Amy and Bob take the best strategy of each one, respectively, determine  $\gcd(A,B)$  when all  $82$  numbers are chosen.

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