
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
Given a natural number \( n \geq 3 \), the numbers \( 1, 2, 3, \ldots, n \) are written on a board. In each move, two numbers are chosen and replaced by their arithmetic mean. This process continues until only one number remains on the board. Determine the smallest integer that can be achieved at the end through an appropriate sequence of moves.

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