
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
Segments \( AA', BB' \), and \( CC' \), each of length 2, all intersect at a point \( O \). If \( \angle AOC' = \angle BOA' = \angle COB' = 60^\circ \), find the maximum possible value of the sum of the areas of triangles \( \triangle AOC', \triangle BOA' \), and \( \triangle COB' \).

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