
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 acute angle \( O \). On one of its sides, we take a point \( A_1 \) and drop a perpendicular \( A_1A_2 \) to the other side of the angle. Then from the point \( A_2 \), we drop a perpendicular to \( OA_1 \), obtaining the point \( A_3 \), and so on. How, using only a compass and straightedge, can we construct a segment equal to the length of the infinite broken line \( L = A_1A_2 + A_2A_3 + \ldots \)? Provide the construction and justify that the resulting segment is of the required length.

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