
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
A regular $\triangle ABC$ has each of its sides divided into four equal parts. Through each division point, lines parallel to the other two sides are drawn. These lines intersect with the sides and the other parallel lines, creating 15 lattice points. If $n$ points are chosen from these 15 lattice points, determine the minimum value of $n$ such that it is guaranteed that three of the chosen points will form an isosceles triangle (including an equilateral triangle). Calculate the minimum value of $n$.

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