
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
Initially, there were $n$ numbers written on the board: $1, 2, \ldots, n$. It is allowed to erase any two numbers on the board and write the absolute value of their difference instead. What is the smallest number that can be on the board after $(n-1)$ such operations:

a) for $n=111$  
b) for $n=110$

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