
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
In Kocourkov, they planned to build a decorative bridge over a river, with its arch being a part of a circle. The profile of the arch, together with the roadway, defines a circular segment. However, in the original design, the height of the bridge arch was too large. Therefore, they built a bridge where the height of the arch was three times smaller, but the radius of the corresponding circle was doubled. What was the ratio of the height of the bridge arch to the radius of the corresponding circle in the design, and what is the ratio for the bridge that was actually built?

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