
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
An equilateral triangle \( \triangle ABC \) has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides of the triangle are drawn. The 15 intersection points of these lines, including the vertices of \( \triangle ABC \), are called lattice points. Out of these 15 lattice points, find the minimum number \( n \) such that among any \( n \) chosen points, there are always three points that form an isosceles triangle (including an equilateral triangle).

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