
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 non-intersecting circles with centers at points \(A\) and \(B\) are inscribed in an angle measuring \(\alpha\) \((0 < \alpha < 180^\circ)\). The distance \(A B\) is \(n\). A segment \(M N\) with its endpoints on the sides of the angle touches both circles. Find the difference in the areas of triangles \(B M N\) and \(A M N\).

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