
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
From point \( A \), rays are drawn that form an angle \(\alpha\) with each other. From an arbitrary point \( B_1 \) on one ray, a perpendicular is drawn to a neighboring ray, and the foot of this perpendicular is labeled \( B_2 \). From this point, another perpendicular is drawn to the next neighboring ray, and this process continues indefinitely. What is the sum of the lengths \( B_1 B_2, B_2 B_3, B_3 B_4 \), etc.? What is the sum of the areas of the triangles \( A B_1 B_2, A B_2 B_3 \), etc., if \( A B_1 = a \)? What must be the value of \(\alpha\) such that the sum of the areas of all the triangles is \( a^2 \)?

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