
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 a scalene triangle \( ABC \), one of the angles is equal to the difference of the other two, and one of the angles is twice another angle. The angle bisectors of \( \angle A \), \( \angle B \), and \( \angle C \) intersect the circumcircle of the triangle at points \( L \), \( O \), and \( M \) respectively. Find the area of triangle \( LOM \) if the area of triangle \( ABC \) is 20. If the answer is not an integer, round it to the nearest whole number.

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