
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
Given an angle \( \angle BAC = \alpha \) with vertex at point \( A \). A perpendicular line \( CD \) is drawn from point \( C \) on \( AC \) to intersect \( AB \). From point \( D \), another perpendicular \( DE \) is drawn to intersect \( AC \), continuing in this fashion infinitely. What is the sum of the lengths of these perpendiculars if \( AC = b \)?

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