
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
Determine the smallest integer \( n \) such that for any configuration of \( n \) points with integer coordinates, there exist two of these points whose midpoint also has integer coordinates.

We recall that if we denote the coordinates of the midpoint between \((x_{1}, y_{1})\) and \((x_{2}, y_{2})\) by \((x_{m}, y_{m})\), then we have \( x_{m} = \frac{x_{1} + x_{2}}{2} \) and \( y_{m} = \frac{y_{1} + y_{2}}{2} \).

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