
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]


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Two touching spheres are inscribed in a dihedral angle of $2\alpha$. Let $A$ be the point of tangency of the first sphere with the first plane, and $B$ be the point of tangency of the second sphere with the second plane. In what ratio does the segment $AB$ get divided by the points of intersection with these spheres?

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