
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 numbers  $ 0$ ,  $ 1$ ,  $ \dots$ ,  $ n$  ( $ n \ge 2$ ) are written on a blackboard. In each step we erase an integer which is the arithmetic mean of two different numbers which are still left on the blackboard. We make such steps until no further integer can be erased. Let  $ g(n)$  be the smallest possible number of integers left on the blackboard at the end. Find  $ g(n)$  for every  $ n$ .

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