
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
In four-dimensional space, the distance between point \( A\left(a_{1}, a_{2}, a_{3}, a_{4}\right) \) and point \( B\left(b_{1}, b_{2}, b_{3}, b_{4}\right) \) is defined as \( AB = \sqrt{\sum_{i=1}^{4}(a_{i} - b_{i})^{2}} \). Consider the set of points
\[ I = \{P(c_{1}, c_{2}, c_{3}, c_{4}) \mid c_{i} = 0 \text{ or } 1, i = 1, 2, 3, 4\} \]

If for any \( n \)-subset \( Q = \{P_{1}, P_{2}, \cdots, P_{n}\} \) of \( I \), there always exists \( P_{i}, P_{j}, P_{k} \in Q \) such that \( \triangle P_{i} P_{j} P_{k} \) is an equilateral triangle, i.e., \( P_{i} P_{j} = P_{j} P_{k} = P_{k} P_{i} \), determine the minimum value of \( n \).

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