
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
For any $n \geq 3$ points $A_{1}$, $A_{2}, \ldots, A_{n}$ in the plane, none of which are collinear, let $\alpha$ be the smallest angle $\angle A_{i} A_{j} A_{k}$ formed by any triple of distinct points $A_{i}, A_{j}, A_{k}$. For each value of $n$, find the maximum value of $\alpha$. Determine the configurations of the points for which this value is attained.

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