
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
The radii of five concentric circles \(\omega_{0}, \omega_{1}, \omega_{2}, \omega_{3}, \omega_{4}\) form a geometric progression with common ratio \(q\). What is the largest value of \(q\) for which it is possible to draw an open broken line \(A_{0} A_{1} A_{2} A_{3} A_{4}\), consisting of four segments of equal length, where \(A_{i}\) lies on \(\omega_{i}\) for all \(i = 0, 1, 2, 3, 4\)?

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