
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 two identical intersecting circles, the ratio of the distance between their centers to the radius is $2m$. A third circle externally touches the first two circles and their common tangent. Find the ratio of the area of the common part of the first two circles to the area of the third circle.

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