
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

As shown in the diagram, $\triangle ABC$ is an isosceles triangle with a vertex angle of $\frac{\pi}{5}$. This triangle is denoted as $\triangle_1$, and its vertex angle is denoted as $\alpha_1 \left( \alpha_1 = \angle BAC \right)$. Construct an isosceles triangle $\triangle_2$ (i.e., $\triangle BCD$) with $\triangle_1$'s base angle $(\angle ABC)$ as its vertex angle, and denote this vertex angle as $\alpha_2 \left( \alpha_2 = \angle DBC \right)$. Then, construct another isosceles triangle $\triangle_3$ (i.e., $\triangle CEB$) with $\triangle_2$'s base angle $(\angle BCD)$ as its vertex angle, and denote this vertex angle as $\alpha_3 \left( \alpha_3 = \angle BCE \right)$. Continue this process to form isosceles triangles $\triangle_1, \triangle_2, \cdots, \triangle_n, \cdots$, with the corresponding vertex angles forming a sequence $\left\{ \alpha_n \right\}$.

(1) Find $\alpha_2, \alpha_3, \alpha_4$.
(2) Find the general term formula for $\alpha_n$.

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