
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
There are 3 integers written on the blackboard. One of the integers is erased and replaced with the difference between the sum of the remaining two integers and 1. This process is repeated several times. Eventually, the numbers on the blackboard become $17, 1967, 1983$. Determine if the initial numbers on the blackboard could have been any of the following sets:
(1) $2, 2, 2$;
(2) $3, 3, 3$.

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