
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
Two spheres with centers \( O_1 \) and \( O_2 \) are intersected by a plane \( P \) that is perpendicular to the segment \( O_1 O_2 \) and passes through its midpoint. The plane \( P \) divides the surface area of the first sphere in the ratio \( m: 1 \) and the surface area of the second sphere in the ratio \( n: 1 \) (where \( m > 1 \) and \( n > 1 \)). Find the ratio of the radii of these spheres.

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